K-Theory and the Enriched Tits Building - uni-bielefeld.de · K-Theory and the Enriched Tits Building To A. A.Suslinwith admiration,on hissixtieth birthday. M. V.Noriand V. Srinivas

Documenta Math. 459

K-Theory and the Enriched Tits Building

To A. A. Suslin with admiration, on his sixtieth birthday.

M. V. Nori and V. Srinivas

Received: December 31, 2009

Revised: May 22, 2010

Abstract. Motivated by the splitting principle, we define certainsimplicial complexes associated to an associative ring A, which havean action of the general linear group GL(A). This leads to an exactsequence, involving Quillen’s algebraic K-groups of A and the symbolmap. Computations in low degrees lead to another view on Suslin’stheorem on the Bloch group, and perhaps show a way towards possiblegeneralizations.

2010 Mathematics Subject Classification: 19D06, 51A05, 55P10Keywords and Phrases: Ki(A), geometries, homotopy types

The homology of GLn(A) has been studied in great depth by A.A. Suslin. Insome of his works ([20] and [21] for example), the action of GLn(A) on certainsimplicial complexes facilitated his homology computations.We introduce three simplicial complexes in this paper. They are motivated bythe splitting principle. The description of these spaces is given below. Thisis followed by the little information we possess on their homology. After thatcomes the connection with K-theory.These objects are defined quickly in the context of affine algebraic groups asfollows. Let G be a connected algebraic group1 defined over a field k. Thecollection of minimal parabolic subgroups P ⊂ G is denoted by FL(G) andthe collection of maximal k-split tori T ⊂ G is denoted by SPL(G). Thesimplicial complex FL(G) has FL(G) as its set of vertices. Minimal parabolicsP0, P1, ..., Pr of G form an r-simplex if their intersection contains a maximalk-split torus2. The dimension of FL(G) is one less than the order of the Weyl

1Gopal Prasad informed us that we should take G reductive or k perfect. The standardclassification, via root systems, of all parabolic subgroups containing a maximal split torus,requires this hypothesis.

2John Rognes has an analogous construction with maximal parabolics replacing minimalparabolics. His spaces, different homotopy types from ours, are connected with K-theory aswell, see [15] .

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460 M. V. Nori and V. Srinivas

group of any T ∈ SPL(G). Dually, we define SPL(G) as the simplicial complexwith SPL(G) as its set of vertices, and T0, T1, ..., Tr forming an r-simplex ifthey are all contained in a minimal parabolic. In general, SPL(G) is infinitedimensional.That both SPL(G) and FL(G) have the same homotopy type can be deducedfrom corollary 7, which is a general principle. A third simplicial complex,denoted by ET(G), which we refer to as the enriched Tits building, is bettersuited for homology computations. This is the simplicial complex whose sim-plices are (nonempty) chains of the partially ordered set E(G) whose definitionfollows. For a parabolic subgroup P ⊂ G, we denote by U(P ) its unipotentradical and by j(P ) : P → P/U(P ) the given morphism. Then E(G) is theset of pairs (P, T ) where P ⊂ G is a parabolic subgroup and T ⊂ P/U(P )is a maximal k-split torus. We say (P ′, T ′) ≤ (P, T ) in E(G) if P ′ ⊂ P andj(P )−1(T ) ⊂ j(P ′)−1T ′. Note that dim E(G) is the split rank of the quotientof G/U(G) by its center. Assume for the moment that this quotient is a simplealgebraic group. Then (P, T ) 7→ P gives a map to the cone of the Tits building.The topology of ET(G) is more complex than the topology of the Tits building,which is well known to be a bouquet of spheres.When G = GL(V ), we denote the above three simplicial complexes by FL(V ),SPL(V ) and ET(V ). These constructions have simple analogues even whenone is working over an arbitrary associative ring A. Their precise definition isgiven with some motivation in section 2. Some basic properties of these spacesare also established in section 2. Amongst them is Proposition 11 which showsthat ET(V ) and FL(V ) have the same homotopy type.ET(An) has a polyhedral decomposition (see lemma 21). This produces aspectral sequence (see Theorem 2) that computes its homology. The E1

p,q terms

and the differentials d1p,q are recognisable since they involve only the homologygroups of ET(Aa) for a < n. The differentials drp,q for r > 1 are not understoodwell enough, however.There are natural inclusions ET(An) → ET(Ad) for d > n, and the inducedmap on homology factors through

Hm(ET(An))→ H0(En(A), Hm(ET(An)))→ Hm(ET(Ad)).

where En(A) is the group of elementary matrices (see Corollary 9).For the remaining statements on the homology of ET(An), we assume that Ais a commutative ring with many units, in the sense of Van der Kallen. See[12] for a nice exposition of the definition and its consequences. Commutativelocal rings A with infinite residue fields are examples of such rings. Under thisassumption, En(A) can be replaced by GLn(A) in the above statement.We have observed that ET(Am+1) has dimensionm. Thus it is natural questionto ask whether

H0(GLm+1(A), Hm(ET(Am+1)))⊗Q→ Hm(ET(Ad))⊗Q

is an isomorphism when d > m + 1. Theorem 3 asserts that this is true form = 1, 2, 3. The statement is true in general (see Proposition 29) if a certain


K-Theory and the Enriched Tits Building 461

Compatible Homotopy Question has an affirmative answer. The higher differ-entials of the spectral sequence can be dealt with if this is true. Proposition 22shows that this holds in some limited situations.The computation of H0(GL3(A), H2(ET(A

3))) ⊗ Q is carried out at in thelast lemma of the paper. This is intimately connected with Suslin’s re-sult (see [21]) connecting K3 and the Bloch group. A closed form forH0(GL4(A), H3(ET(A

4)) is awaited. This should impact on the study ofK4(A).We now come to the connection with the Quillen K-groups Ki(A) as obtainedby his plus construction.GL(A) acts on the geometric realisation |SPL(A∞)| and thus we have theBorel construction, namely the quotient of |SPL(A∞)|×EGL(A) by GL(A), afamiliar object in the study of equivariant homotopy. We denote this space bySPL(A∞)//GL(A). We apply Quillen’s plus construction to SPL(A∞)//GL(A)and a suitable perfect subgroup of its fundamental group to obtain a spaceY (A). Proposition 17 shows that Y (A) is an H-space and that the natural mapY (A) → BGL(A)+ is an H-map. Its homotopy fiber, denoted by SPL(A∞)+,is thus also a H-space. The n-th homotopy group of SPL(A∞)+ at its canonicalbase point is denoted by Ln(A). There is of course a natural map SPL(A∞)→SPL(A∞)+. That this map is a homology isomorphism is shown in lemma 16.This assertion is easy, but not tautological: it relies once again on the trivialityof the action of E(A) on H∗(SPL(A

∞)). As a consequence of this lemma,Ln(A) ⊗ Q is identified with the primitive rational homology of SPL(A∞), orequivalently, that of ET(A∞).We have the inclusion Nn(A) → GLn(A), where Nn(A) is the semidirect prod-uct of the permutation group Sn with (A×)n. Taking direct limits over n ∈ N,we obtain N(A) ⊂ GL(A). Let H ′ ⊂ N(A) be the infinite alternating groupand let H be the normal subgroup generated by H ′. Applying Quillen’s plusconstruction to the space BN(A) with respect to H , we obtain BN(A)+. Itsn-th homotopy group is defined to be Hn(A

×). From the Dold-Thom theo-rem, it is easy to see that Hn(A

×) ⊗ Q is isomorphic to the group homologyHn(A

×) ⊗ Q. When A is commutative, this is simply ∧nQ(A× ⊗ Q). Proposi-

tion 20 identifies the groups Hn(A×) with certain stable homotopy groups. Its

proof was shown to us by J. Peter May. It is sketched in the text of the paperafter the proof of the Theorem below.

Theorem 1. Let A be a Nesterenko-Suslin ring. Then there is a long exactsequence, functorial in A:

· · · → L2(A) → H2(A×) → K2(A) → L1(A) → H1(A

×) → K1(A) → L0(A) → 0.

We call a ring A Nesterenko-Suslin if it satisfies the hypothesis of Remark 1.13of their paper [13]. The precise requirement is that for every finite set F , thereis a function fF : F → the center of A so that the sum ΣfF (s) : s ∈ S is aunit of A for every nonempty S ⊂ F . If k is an infinite field, every associativek-algebra is Nesterenko-Suslin, and so is every commutative ring with manyunits in the sense of Van der Kallen. Remark 1.13 of Nesterenko-Suslin [13]



permits us to ignore unipotent radicals. This is used crucially in the proof ofTheorem 1 (see also Proposition 12).In the first draft of the paper, we conjectured that this theorem is true withoutany hypothesis on A. Sasha Beilinson then brought to our attention Suslin’s pa-per [22] on the equivalence of Volodin’s K-groups and Quillen’s. From Suslin’sdescription of Volodin’s spaces, it is possible to show that these spaces are ho-motopy equivalent to the total space of the Nn(A)-torsor on FL(An) given insection 2 of this paper. This requires proposition 1 and a little organisation.Once this is done, Corollary 9 can also be obtained from Suslin’s set-up. Thestatement “X(R) is acyclic” stated and proved by Suslin in [22] now validatesProposition 12 at the infinite level, thus showing that Theorem 1 is true withoutany hypothesis on A. The details have not been included here.R. Kottwitz informed us that the maximal simplices of FL(V ) are referred toas “regular stars” in the work of Langlands( see [5]).We hope that this paper will eventually connect with mixed Tate motives(see[3],[1]).The arrangement of the paper is a follows. Section 1 has some topologicalpreliminaries used through most of the paper. The proofs of Corollary 7 andProposition 11 rely on Quillen’s Theorem A. Alternatively, they can both beproved directly by repeated applications of Proposition 1. The definitions ofSPL(An),FL(An),ET(An) and first properties are given in section two. Thenext four sections are devoted to the proof of Theorem 1. The last four sectionsare concerned with the homology of ET(An).The lemmas, corollaries and propositions are labelled sequentially. For in-stance, corollary 9 is followed by lemma 10 and later by proposition 11; thereis no proposition 10 or corollary 10. The other numbered statements are thethree theorems. Theorems 2 and 3 are stated and proved in sections 7 and 9respectively. Section 0 records some assumptions and notation, some perhapsnon-standard, that are used in the paper. The reader might find it helpful toglance at this section for notation regarding elementary matrices and the Borelconstruction and the use of “simplicial complexes”.

0. Assumptions and Notation

Rings, Elementary matrices, Elem(W → V ),Elem(V, q), L(V ) and Lp(V )

We are concerned with the Quillen K-groups of a ring A.We assume that A has the following property: if Am ∼= An as left A-modules,then m = n. The phrase “A-module” always means left A-module.For a finitely generated free A-module V , the collection of A-submodules L ⊂ Vso that (i) V/L is free and (ii) L is free of rank one , is denoted by L(V ).Lp(V ) is the collection of subsets q of cardinality (p+ 1) of L(V ) so that⊕L : L ∈ q → V is a monomorphism whose cokernel is free.Given an A-submodule W of a A-module V so that the short exact sequence

0→W → V → V/W → 0



is split, we have the subgroup Elem(W → V ) ⊂ AutA(V ), defined as follows.Let H(W ) be the group of automorphisms h of V so that(idV − h)V ⊂ W ⊂ ker(idV − h). Let W ′ ⊂ V be a submodule that iscomplementary to W . Define H(W ′) in the same manner. The subgroup ofAutA(V ) generated byH(W ) andH(W ′) is Elem(W → V ). It does not dependon the choice of W ′ because H(W ) acts transitively on the collection of suchW ′.For example, if V = An, and W is the A-submodule generated by any r mem-bers of the given basis of An, then Elem(W → An) equals En(A) , the subgroupof elementary matrices in GLn(A), provided of course that 0 < r < n.If V is finitely generated free and if q ∈ Lp(V ), the above statement impliesthat Elem(L → V ) does not depend on the choice of L ∈ q. Thus we denotethis subgroup by Elem(V, q) ⊂ GL(V ).

The Borel Construction

Let X be a topological space equipped with the action of group G. Let EGbe the principal G-bundle on BG (as in [14]). The Borel construction, namelythe quotient of X × EG by the G-action, is denoted by X//G throughout thepaper.

Categories, Geometric realisations, Posets

Every category C gives rise to a simplicial set, namely its nerve (see [14]). Itsgeometric realisation is denoted by BC.A poset (partially ordered set) P gives rise to a category. The B-constructionof this category, by abuse of notation, is denoted by BP . Associated to Pis the simplicial complex with P as its set of vertices; the simplices are finitenon-empty chains in P . The geometric realisation of this simplicial complexcoincides with BP .

Simplicial Complexes, Products and Internal Hom, Barycentric subdivision

Simplicial complexes crop up throughout this paper. We refer to Chapter 3,[18],for the definition of a simplicial complex and its barycentric subdivision. S(K)and V(K) denote the sets of vertices and simplices respectively of a simplicialcomplex K. The geometric realisation of K is denoted by |K|. The set S(K) isa partially ordered set (with respect to inclusion of subsets). Note that BS(K)is simply the (geometric realisation of) the barycentric subdivision sd(K). Thegeometric realisations of K and sd(K) are canonically homeomorphic to eachother, but not by a simplicial map.Given simplicial complexes K1 and K2, the product |K1| × |K2| (in the com-pactly generated topology) is canonically homeomorphic to B(S(K1)×S(K2)).

The category of simplicial complexes and simplicial maps has a categori-cal product :V(K1×K2) = V(K1)×V(K2). A non-empty subset of V(K1×K2) is a simplexof K1 × K2 if and only if it is contained in S1 × S2 for some Si ∈ S(Ki) fori = 1, 2. The geometric realisation of the product is not homeomorphic to theproduct of the geometric realisations, but they do have the same homotopy



type. In fact Proposition 1 of section 1 provides a contractible collection ofhomotopy equivalences |K1|× |K2| → |K1×K2|. For most purposes, it sufficesto note that there is a canonical map P (K1,K2) : |K1| × |K2| → |K1 × K2|.This is obtained in the following manner. Let C(K) denote the R-vector spacewith basis V(K) for a simplicial complex K. Recall that |K| is a subset ofC(K). For simplicial complexes K1 and K2, we have the evident isomorphism

j : C(K1)⊗R C(K2)→ C(K1 ×K2).

For ci ∈ |Ki| for i = 1, 2 we put P (K1,K2)(c1, c2) = j(c1 ⊗ c2) ∈ C(K1 ×K2).We note that j(c1 ⊗ c2) belongs to the subset |K1 × K2|. This gives thecanonical P (K1,K2).

Given simplicial complexes K,L there is a simplicial complex Hom(K,L)with the following property: if M is a simplicial complex, then the set ofsimplicial maps K ×M → L is naturally identified with the set of simplicialmaps M → Hom(K,L). This simple verification is left to the reader.

Simplicial maps f : K1 × K2 → K3 occur in sections 2 and 5 of thispaper.

|f | P (K1,K2) : |K1| × |K2| → |K3|

is the map we employ on geometric realisations. Maps |K1| × |K2| → |K3|associated to simplicial maps f1 and f2 are seen (by contiguity) to be homotopicto each other if f1, f2 is a simplex ofHom(K1×K2,K3). This fact is employedin Lemma 8.Simplicial maps f : K1×K2 → K3 are in reality maps V(f) : V(K1)×V(K2)→V(K3) with the property that V(f)(S1 × S2) is a simplex of K3 whenever S1

and S2 are simplices of K1 and K2 respectively. One should note that suchan f induces a map of posets S(K1)× S(K2)→ S(K3), which in turn inducesa continuous map B(S(K1) × S(K2)) → BS(K3). In view of the naturalidentifications, this is the same as giving a map |K1| × |K2| → |K3|. This mapcoincides with the |f | P (K1,K2) considered above.The homotopy assertion of maps |K1| × |K2| → |K3| associated to f1, f2 wheref1, f2 is an edge of Hom(K1 ×K2,K3) cannot be proved by the quick posetdefinition of the maps (for |K3| has been subdivided and contiguity is notavailable any more). This explains our preference for the longwinded |f | P (K1,K2) definition.

1. Some preliminaries from topology

We work with the category of compactly generated weakly Hausdorff spaces.A good source is Chapter 5 of [11]. This category possesses products. Italso possesses an internal Hom in the following sense: for compactly generatedHausdorff X,Y, Z, continuous maps Z → Hom(X,Y ) are the same as continu-ous maps Z×X → Y , where Z×X denotes the product in this category. Thisinternal Hom property is required in the proof of Proposition 1 stated below.



Hom(X,Y ) is the space of continuous maps from X to Y . This space ofmaps has the compact-open topology, which is then replaced by the inheritedcompactly generated topology. This space Hom(X,Y ) is referred to frequentlyas Map(X,Y ), and some times even as Maps(X,Y ) , in the text.Now consider the following set-up. Let Λ be a partially ordered set assumedto be Artinian: (i) every non-empty subset in Λ has a minimal element withrespect to the partial order, or equivalently (ii) there are no infinite strictly de-scending chains λ1 > λ2 > · · · in Λ. The poset Λ will remain fixed throughoutthe discussion below.We consider topological spaces X equipped with a family of closed subsetsXλ, λ ∈ Λ with the property that Xµ ⊂ Xλ whenever µ ≤ λ.Given another Y, Yλ, λ ∈ Λ as above, the collection of Λ-compatible contin-uous f : X → Y (i.e. satisfying f(Xλ) ⊂ Yλ, ∀λ ∈ Λ) will be denoted byMapΛ(X,Y ). MapΛ(X,Y ) is a closed subset of Hom(X,Y ), and this topolo-gises MapΛ(X,Y ).We say that Xλ is a weakly admissible covering of X if the three conditionslisted below are satisfied. It is an admissible covering if in addition, each Xλ

is contractible.

(1) For each pair of indices λ, µ ∈ Λ, we have

Xλ ∩Xµ = ∪ν≤λ,ν≤µ

Xν

(2) If∂Xλ = ∪

ν<λXν ,

then ∂Xλ → Xλ is a cofibration(3) The topology on X is coherent with respect to the family of subsetsXλλ∈Λ, that is, X = ∪λXλ, and a subset Z ⊂ X is closed preciselywhen Z ∩Xλ is closed in Xλ in the relative topology, for all λ.

Proposition 1. Assume that Xλ is a weakly admissible covering of X. As-sume also that each Yλ is contractible.Then the space MapΛ(X,Y ) of Λ-compatible maps f : X → Y is contractible.In particular, it is non-empty and path-connected.

Corollary 2. If both Xλ and Yλ are admissible, then X and Y arehomotopy equivalent.

With assumptions as in the above corollary, the proposition yields the existenceof Λ-compatible maps f : X → Y and g : Y → X . Because g f and f gare also Λ-compatible, that they are homotopic to idX and idY respectively isdeduced from the path-connectivity of MapΛ(X,X) and MapΛ(Y, Y ).

Corollary 3. If Xλ is admissible, then there is a homotopy equivalenceX → BΛ.

Here, recall that BΛ is the geometric realization of the simplicial complexassociated to the set of nonempty finite chains (totally ordered subsets) in Λ;equivalently, regarding Λ as a category, BΛ is the geometric realization of its



nerve. We put Y = BΛ and Yλ = Bµ ∈ Λ : µ ≤ λ in Corollary 2 to deduceCorollary 3.The proof of Proposition 1 is easily reduced to the following extension lemma.

Lemma 4. Let Xλ, Yλ etc. be as in the above proposition. Let Λ′ ⊂ Λ be asubset, with induced partial order, so that for any λ ∈ Λ′, µ ∈ Λ with µ ≤ λ, wehave µ ∈ Λ′. Let X ′ = ∪λ∈Λ′Xλ, Y

′ = ∪λ∈Λ′Yλ. Assume given a continuousmap f ′ : X ′ → Y ′ with f ′(Xλ) ⊂ Yλ for all λ ∈ Λ′. Then f ′ extends to acontinuous map f : X → Y with f(Xλ) ⊂ Yλ for all λ ∈ Λ.

Proof. Consider the collection of pairs (Λ′′, f ′′) satisfying:(a) Λ′ ⊂ Λ′′ ⊂ Λ(b) µ ∈ Λ, λ ∈ Λ′′, µ ≤ λ implies µ ∈ Λ′′

(c) f ′′ : ∪Xµ|µ ∈ Λ′′ → Y is a continuous map(d) f ′′(Xµ) ⊂ Yµ for all µ ∈ Λ′′

(e) f ′|Xµ = f ′′|Xµ for all µ ∈ Λ′

This collection is partially ordered in a natural manner. The coherence con-dition on the topology of X ensures that every chain in this collection has anupper bound. The presence of (Λ′, f ′) shows that it is non-empty. By Zorn’slemma, there is a maximal element (Λ′′, f ′′) in this collection. The Artinianhypothesis on Λ shows that if Λ′′ 6= Λ, then its complement possesses a mini-mal element µ. Let D′′ be the domain of f ′′. The minimality of µ shows thatD′′ ∩Xµ = ∂Xµ. By condition (d) above, we see that f ′′(∂Xµ) is contained inthe contractible space Yµ. Because ∂Xµ → Xµ is a cofibration, it follows thatf ′′|∂Xµ extends to a map g : Xµ → Yµ. The f ′′ and g patch together to givea continuous map h : D′′ ∪ Xµ → Y . Since the pair (Λ′′ ∪ µ, h) evidentlybelongs to this collection, the maximality of (Λ′′, f ′′) is contradicted. ThusΛ′′ = Λ and this completes the proof.

The proof of the Proposition follows in three standard steps.Step 1: Taking Λ′ = ∅ in Lemma 4 we deduce that MapΛ(X,Y ) is nonempty.Step 2: For the path-connectivity of MapΛ(X,Y ) , we replace X by X × [0, 1]and replace the original poset Λ by the product Λ×0, 1, 0, 1, with theproduct partial order, where the second factor is partially ordered by inclusion.The subsets of X × I (resp.Y ) indexed by (λ, 0), (λ, 1), (λ, 0, 1) are Xλ ×0, Xλ × 1 and Xλ × [0, 1] (resp. Yλ in all three cases).We then apply the lemma to the sub-poset Λ× 0, 1.Step 3: Finally, for the contractibility of MapΛ(X,Y ), we first choose f0 ∈MapΛ(X,Y ) and then consider the two maps MapΛ(X,Y ) × X → Y givenby (f, x) 7→ f(x) and (f, x) 7→ f0(x). Putting (MapΛ(X,Y ) × X)λ =MapΛ(X,Y ) × Xλ for all λ ∈ Λ, we see that both the above maps are Λ-compatible. The path-connectivity assertion in Step 2 now gives a homotopybetween the identity map of MapΛ(X,Y ) and the constant map f 7→ f0. Thiscompletes the proof of Proposition 1.We now want to make some remarks about equivariant versions of the abovestatements.



Given X, Xλ;λ ∈ Λ as above, an action of a group G on X is called Λ-compatible if G also acts on the poset Λ so that for all g ∈ G, λ ∈ Λ, we haveg(Xλ) = Xgλ.Under the conditions of Proposition 1, suppose Xλ, Yλ admit Λ-compatibleG-actions. There is no G-equivariant f ∈ MapΛ(X,Y ) in general. However,if f ∈ MapΛ(X,Y ) and gX , gY denote the actions of g ∈ G on X and Yrespectively, we see that , then gY f g

−1X is also a Λ-compatible map. By

Proposition 1, we see that this map is homotopic to f . Thus gY f and f gXare homotopic to each other. In particular, Hn(f) : Hn(X) → Hn(Y ) is ahomomorphism of G-modules.In the sequel a better version of this involving the Borel construction is needed.We recall the Borel construction of equivariant homotopy quotient spaces. LetEG denote a contractible CW complex on which G has a proper free cellularaction; for our purposes, it suffices to fix a choice of this space EG to be thegeometric realization of the nerve of the translation category of G (the categorywith vertices [g] indexed by the elements of G, and unique morphisms betweenordered pairs of vertices ([g], [h]), thought of as given by the left action of hg−1).The classifying space BG is the quotient space EG/G.If X is any G-space, let X//G denote the homotopy quotient of X by G, ob-tained using the Borel construction, i.e.,

(1) X//G = (X × EG)/G,

where EG is as above, and G acts diagonally. Note that the natural quotientmap

qX : X × EG→ X//G

is a Galois covering space, with covering group G.

If X and Y are G-spaces, then considering G-equivariant maps f : X ×EG→Y ×EG compatible with the projections to EG, giving a commutative diagram

X × EG

$$JJJJJJJJJ

f// Y × EG

zzttttttttt

EG

is equivalent to considering maps f : X//G→ Y//G compatible with the mapsqX : X//G→ BG, qY : Y//G→ BG, giving a commutative diagram

X//G

qX##GGGGGGGG

f// Y//G

qYwwwwwwww

BG

Proposition 5. Assume that, in the situation of proposition 1, there are Λ-compatible G-actions on X and Y . Let EG be as above, and consider the Λ-compatible families Xλ × EG, which is a weakly admissible covering familyfor X×EG, and Yλ×EG, which is an admissible covering family for Y ×EG.



Then there is a G-equivariant map f : X × EG → Y × EG, compatible withthe projections to EG, such that

(i) f(Xλ × EG) ⊂ (Yλ × EG) for all λ ∈ Λ(ii) if g : X × EG → Y × EG is another such equivariant map, then

there is a G-equivariant homotopy between f and g, compatible withthe projections to EG

(iii) The space of such equivariant maps X × EG→ Y × EG, as in (i), iscontractible.

Proof. We show the existence of the desired map, and leave the proof of otherproperties, by similar arguments, to the reader.Let MapΛ(X,Y ) be the contractible space of Λ-compatible maps from X toY ; note that it comes equipped with a natural G-action, so that the canon-ical evaluation map X × MapΛ(X,Y ) → Y is equivariant. This inducesX × MapΛ(X,Y ) × EG → Y × EG. There is also a natural G-equivariantmap π : X × MapΛ(X,Y ) × EG → X × EG. This map π has equivariantsections, since the projection MapΛ(X,Y )×EG→ EG is a G-equivariant mapbetween weakly contractible spaces, so that the map on quotients modulo G isa weak homotopy equivalence (i.e., (MapΛ(X,Y )×EG)/G is another “model”for the classifying space BG = EG/G). However BG is a CW complex, so themap has a section.

As another preliminary, we note some facts (see lemma 6 below) which areessentially corollaries of Quillen’s Theorem A (these are presumably well-knownto experts, though we do not have a specific reference).If P is any poset, let C(P ) be the poset consisting of non-empty finite chains(totally ordered subsets) of P . If f : P → Q is a morphism between posets (anorder preserving map) there is an induced morphism C(f) : C(P ) → C(Q).If S is a simplicial complex (literally, a collection of finite non-empty subsetsof the vertex set), we may regard S as a poset, partially ordered with respectto inclusion; then the classifying space BS is naturally homeomorphic to thegeometric realisation |S| (and gives the barycentric subdivision of |S|). A sim-plicial map f : S → T between simplicial complexes (that is, a map on vertexsets which sends simplices to simplices, not necessarily preserving dimension)is also then a morphism of posets. We say that a poset P is contractible if itsclassifying space BP is contractible.

Lemma 6. (i) Let f : P → Q be a morphism between posets. Supposethat for each X ∈ C(Q), the fiber poset C(f)−1(X) is contractible. ThenBf : BP → BQ is a homotopy equivalence.(ii) Let f : S → T be a simplicial map between simplicial complexes. Sup-pose that for any simplex σ ∈ T , the fiber f−1(σ), considered as a poset, iscontractible. Then |f | : |S| → |T | is a homotopy equivalence.

Proof. We first prove (i). For any poset P , there is morphism of posets ϕP :C(P ) → P , sending a chain to its first (smallest) element. If a, b ∈ P witha ≤ b, and C is a chain in ϕ−1

P (b), then a ∪ C is a chain in ϕ−1P (a). This



gives an order preserving map of posets ϕ−1P (b)→ ϕ−1

P (a) (i.e., a “base-change”functor). This makes C(P ) prefibred over P , in the sense of Quillen (see page96 in [19], for example). Also, ϕ−1

P (a) has the minimal element (initial object)a, and so its classifying space is contractible.Hence Quillen’s Theorem A (see [19], page 96) implies that B(ϕP ) is a homo-topy equivalence, for any P .Now let f : P → Q be a morphism between posets. Let C(f) : C(P )→ C(Q)be the corresponding morphism on the posets of (finite, nonempty) chains.If A ⊂ B are two chains in C(Q), there is an obvious order preserving mapC(f)−1(B)→ C(f)−1(A). Again, this makes C(f) : C(P )→ C(Q) prefibred.Since we assumed that BC(f)−1(A) is contractible, for all A ∈ C(Q), Quillen’sTheorem A implies that BC(f) is a homotopy equivalence.We thus have a commutative diagram of posets and order preserving maps

C(P )C(f)→ C(Q)

ϕP ↓ ↓ ϕQ

Pf→ Q

where three of the four sides yield homotopy equivalences on passing to clas-sifying spaces. Hence Bf : BP → BQ is a homotopy equivalence, proving(i).The proof of (ii) is similar. This is equivalent to showing that Bf : BS → BTis a homotopy equivalence. Since f : S → T , regarded as a morphism of posets,is naturally prefibered, and by assumption, Bf−1(σ) is contractible for eachσ ∈ T , Quillen’s Theorem A implies that Bf is a homotopy equivalence.

We make use of Propositions 1 and 5 in the following way.Let A, B be sets, Z ⊂ A × B a subset such that the projections p : Z → A,q : Z → B are both surjective. Consider simplicial complexes SZ(A), SZ(B)on vertex sets A, B respectively, with simplices in SZ(A) being finite nonemptysubsets of fibers q−1(b), for any b ∈ B, and simplices in SZ(B) being finite,nonempty subsets of fibers p−1(a), for any a ∈ A.Consider also a third simplicial complex SZ(A,B) with vertex set Z, where afinite non-empty subset Z ′ ⊂ Z is a simplex if and only it satisfies the followingcondition:

(a1, b1), (a2, b2) ∈ Z ′ ⇒ (a1, b2) ∈ Z.

Note that the natural maps on vertex sets p : Z → A, q : Z → B inducecanonical simplicial maps on geometric realizations

|p| : |SZ(A,B)| → |SZ(A)|, |q| : |SZ(A,B)| → |SZ(B)|.

Corollary 7. (1) With the above notation, the simplicial maps

|p| : |SZ(A,B)| → |SZ(A)|, |q| : |SZ(A,B)| → |SZ(B)|

are homotopy equivalences. In particular, |SZ(A)|, |SZ(B)| are homo-topy equivalent.



(2) If a group G acts on A and on B, so that Z is stable under the diagonalG action on A × B, then the homotopy equivalences |p|, |q| are G-equivariant homotopy equivalences. Hence there exists a natural G-equivariant homotopy equivalence between |SZ(A)|×EG and |SZ(B)|×EG.

Proof. Since the situation is symmetric with respect to the sets A, B, it sufficesto show |p| is a homotopy equivalence. Note that in the context of a G-actionas stated, the G-equivariance of |p| is clear.Let Λ be the poset of all simplices of SZ(A), thought of as subsets of A, andordered by inclusion. Clearly Λ is Artinian.Apply Corollary 2 with X = |SZ(A,B)|, Y = |SZ(A)|, Λ as above, and thefollowing Λ-admissible coverings: for σ ∈ Λ, let Yσ be the (closed) simplex inY = |SZ(A)| determined by σ (clearly Yσ is admissible); takeXσ = |p|−1(Yσ)(this is evidently weakly admissible). For admissibility of Xσ, we need toshow that each Xσ is contractible.In fact, regarding the sets of simplices SZ(A,B) and SZ(A) as posets, andSZ(A,B) → SZ(A) as a morphism of posets, Xσ is the geometric realizationof the simplicial complex determined by ∪τ≤σp

−1(τ).The corresponding map of posets

p−1(τ |τ ≤ σ)→ τ |τ ≤ σ

has contractible fiber posets – if we fix an element x ∈ p−1(τ), and p−1(τ)(≥ x)is the sub-poset of elements bounded below by x, then y 7→ y∪x is a morphismof posets rx : p−1(τ) → p−1(τ)(≥ x) which gives a homotopy equivalence ongeometric realizations (it is left adjoint to the inclusion of the sub-poset). Butthe sub-poset has a minimal element, and so its realization is contractible.The poset τ |τ ≤ σ is obviously contractible, since it has a maximal element.Hence, applying lemma 6(ii), Xσ is contractible.Since the map |p| : X → Y is Λ-compatible, it is the unique such map uptoΛ-compatible homotopy, and is a homotopy equivalence.

We note that the argument with Quillen’s Theorem A in fact implies directlythat SZ(A,B) → SZ(A) is a homotopy equivalence; the uniqueness assertionis not, apparently, a formal consequence of Quillen’s Theorem A.

2. Flag Spaces

In this section, we discuss various constructions of spaces (generally simplicialcomplexes) defined using flags of free modules, and various maps, and homotopyequivalences, between these. These are used as building blocks in the proof ofTheorem 1.Let A be a ring, and let V be a free (left) A-module of rank n. Define asimplicial complex FL(V ) as follows.Its vertex set is

FL(V ) =



= F = (F0, F1, . . . , Fn) | 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fn = V are A-submodules,

and each quotient Fi/Fi−1 is a free A-module of rank 1.

We think of this vertex set as the set of “full flags” in V .To describe the simplices in FL(V ), we need another definition. Let

SPL(V ) = L1, . . . , Ln | Li ⊂ V is a free A-submodule of rank 1, and

the induced map ⊕ni=1Li → V is an isomorphism.

Note that L1, . . . , Ln is regarded as an unordered set of free A-submodules ofrank 1 of L (i.e., as a subset of cardinality n in the set of all free A-submodulesof rank 1 of V ). We think of SPL(V ) as the “set of unordered splittings of Vinto direct sums of free rank 1 modules”.Given α ∈ SPL(V ), say α = L1, . . . , Ln, we may choose some ordering(L1, . . . , Ln) of its elements, and thus obtain a full flag in V (i.e., an elementin FL(V )), given by

(0, L1, L1 ⊕ L2, · · · , L1 ⊕ · · · ⊕ Ln = V ) ∈ FL(V ).

Let

[α] ⊂ FL(V )

be the set of n! such full flags obtained from α.We now define a simplex in FL(V ) to be any subset of such a set [α] of vertices,for any α ∈ SPL(V ). Thus, FL(V ) becomes a simplicial complex of dimen-sion n! − 1, with the sets [α] as above corresponding to maximal dimensionalsimplices.Clearly Aut (V ) ∼= GL n(A) acts on the simplical complex FL(V ) through sim-plicial automorphisms, and thus acts on the homology groups H∗(FL(V ),Z)(and other similar invariants of FL(V )).Next, remark that if F ∈ FL(V ) is any vertex of FL(V ), we may associateto it the free A-module gr F (V ) = ⊕n

i=1Fi/Fi−1. If (F, F ′) is an ordered pairof distinct vertices, which are joined by an edge in FL(V ), then we obtain acanonical isomorphism (determined by the edge)

ϕF,F ′ : gr F (V )∼=−→ gr F ′(V ).

One way to describe it is by considering the edge as lying in a simplex [α],for some α = L1, . . . , Ln ∈ SPL(V ); this determines an identification ofgr F (V ) with ⊕iLi, and a similar identification of gr F ′(V ), and thereby anidentification between grF (V ) and grF ′(V ). Note that from this descriptionof the maps ϕF.F ′ , it follows that if F, F ′, F ′′ form vertices of a 2-simplex inFL(V ), i.e., there exists some α ∈ SPL(V ) such that F, F ′, F ′′ ∈ [α], then wealso have

ϕF,F ′′ = ϕF ′,F ′′ ϕF,F ′ .

The isomorphism ϕF,F ′ depends only on the (oriented) edge in FL(V ) deter-mined by (F, F ′), and not on the choice of the simplex [α] in which it lies. Oneway to see this is to use that, for any two such filtrations F , F ′ of V there is acanonical isomorphism gr pF gr

qF ′(V ) ∼= gr qF ′gr

pF (V ) (Schur-Zassenhaus lemma)



for each p, q. But in case F , F ′ are flags which are connected by an edge,then there is also a canonical isomorphism gr F gr F ′(V ) ∼= gr F ′(V ) (in fact theF -filtration induced on gr pF ′(V ) has only 1 non-trivial step, for each p), andsimilarly there is a canonical isomorphism gr F ′grF (V ) ∼= gr F (V ). These threecanonical isomorphisms combine to give the isomorphism ϕF,F ′ .Hence there is a well-defined local system gr(V ) of A-modules on the geometricrealization |FL(V )| of the simplicial complex FL(V ), whose fibre over a vertexF is gr F (V ).Notice further that this local system gr(V ) comes equipped with a naturalAut(V ) action, compatible with the natural actions on FL(V ) and FL(V ).Indeed, any element g ∈ Aut(V ) gives a bijection on the set of full flags FL(V ),with

F = (0 = F0, F1, . . . , Fn = V ) ∈ FL(V ))

mapping to

gF = (0 = gF0, gF1, . . . , gFn = V ).

This clearly gives an induced isomorphism ⊕iFi/Fi−1∼= ⊕igFi/gFi−1, identi-

fying the fibers of the local system over F and gF in a specific way. It is easy tosee that if α = L1, . . . , Ln ∈ SPL(V ), then gα = gL1, . . . , gLn ∈ SPL(V ),giving the action of Aut (V ) on SPL(V ), so that if a pair of vertices F, F ′ ofFL(V ) lie on an edge contained in [α], then gF, gF ′ lie on an edge contained in[gα], and so the induced identification ϕF,F ′ is compatible with ϕgF,gF ′ . Thisinduces the desired action of Aut(V ) on the local system.Further, note that if F, F ′ ∈ FL(V ) are connected by an edge in FL(V ), then wemay realize ϕF,F ′ by the action of a suitable element of Aut (V ), which preservesa simplex [α] in which the edge lies, and permutes the lines in the splittingα ∈ SPL(V ). Thus, given any edge-path joining vertices F, F ′ in FL(V ), theinduced composite isomorphism gr F (V ) → gr F ′(V ) is again realized by theaction of an element of Aut (V ). In particular, given an edge-path loop basedas F ∈ FL(V ), the induced automorphism of gr F (V ) is induced by the actionon gr F (V ) of an element of the isotropy group of F in Aut (V ), which is the“Borel subgroup” corresponding to the flag F .Hence, the monodromy group of the local system gr(V ) is clearly contained inNn(A), defined as a semidirect product

(2) Nn(A) = (A× × · · · ×A×)⋉ Sn

where Sn is the permutation group; we regard Nn(A) as a subgroup ofAut (⊕iLi) in an obvious way.Now we make infinite versions of the above constructions.Let A∞ be the set of sequences (a1, a2, . . . , an, . . .) of elements of A, all butfinitely many of which are 0, considered as a free A-module of countable rank.There is a standard inclusion in : An → A∞ of the standard free A-moduleof rank n as the submodule of sequences with am = 0 for all m > n. Theinduced inclusion i : An → An+1 is the usual one, given by i(a1, . . . , an) =(a1, . . . , an, 0).



We may thus view A∞ as being given with a tautological flag, consisting of theA-submodules in(A

n). We define a simplicial complex FL(A∞), with vertexset FL(A∞) equal to the set of flags 0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ A∞ whereVi/Vi−1 is a free A-module of rank 1, for each i ≥ 1, and with Vn = i(An) forall sufficiently large n. Thus FL(A∞) is naturally the union of subsets bijectivewith FL(An). To make FL(A∞) into a simplicial complex, we define a simplexto be a finite set of vertices in some subset FL(An) which determines a simplexin the simplicial complex FL(An); this property does not depend on the choiceof n, since the natural inclusion FL(An) → FL(An+1), regarded as a map onvertex sets, identifies FL(An) with a subcomplex of FL(An+1), such that anysimplex of FL(An+1) with vertices in FL(An) is already in the subcomplexFL(An).We consider GL(A) ⊂ Aut(A∞) as the union of the images of the obviousmaps in : GLn(A) → Aut (A∞), obtained by automorphisms which fix allthe basis elements of A∞ beyond the first n. We clearly have an induced ac-tion of GL(A) on the simplicial complex FL(A∞), and hence on its geometricrealisation |FL(A∞)| through homeomorphisms preserving the simplicial struc-ture. The inclusion FL(An) → FL(A∞) as a subcomplex is clearly GLn(A)-equivariant.Next, observe that there is a local system gr(A∞) on FL(A∞) whose fiberover a vertex F = (F0 = 0, F1, . . . , Fn, . . . ) is grF (V ) = ⊕iFi/Fi−1. This hasmonodromy contained in

N(A) = ∪nNn(A) ⊂ GL(A),

where we may also view N(A) as the semidirect product of

(A×)∞ = diagonal matrices in GL(A)

by the infinite permutation group S∞. This local system also carries a naturalGL(A)-action, compatible with the GL(A)-action on FL(A∞).Next, we prove a property (Corollary 9) about the action of elementary matriceson homology, which is needed later. The corollary follows immediately fromthe lemma below.For the statement and proof of the lemma, we suggest that the reader browsethe remarks on Hom(K1×K2,K3) in section 0, given simplicial complexes Ki

for i = 1, 2, 3. The notation Elem(V ′ → V ′ ⊕ V ′′) that appears in the lemmahas also been introduced in section 0 under the heading “elementary matrices”.

Lemma 8. Let V ′, V ′′ be two free A-modules of finite rank, i′ : V ′ → V ′⊕V ′′,i′′ : V ′′ → V ′ ⊕ V ′′ the inclusions of the direct summands. Consider the twonatural maps

(3) α, β : FL(V ′)× FL(V ′′)→ FL(V ′ ⊕ V ′′)

given by

α : ((F ′1, . . . , F

′r = V ′), (F ′′

1 , . . . , F′′s )) 7→

(i′(F ′1), . . . , i

′(F ′r) = i′(V ′), i′(V ′) + i′′(F ′′

1 ), . . . , i′(V ′) + i′′(F ′′

s ) = V ′ ⊕ V ′′),



β : ((F ′1, . . . , F

′r = V ′), (F ′′

1 , . . . , F′′s )) 7→

(i′′(F ′′1 ), . . . , i

′′(F ′′s ) = i′′(V ′′), i′′(V ′′) + i′(F ′

1), . . . , i′′(V ′′) + i′(F ′

r) = V ′⊕ V ′′).

(A) α and β are vertices of a one-simplex of Hom(FL(V ′)× FL(V ′′),FL(V ′ ⊕V ′′)).(B) The maps |FL(V ′)| × |FL(V ′′)| → |FL(V ′ ⊕ V ′′)| induced by α, β are ho-motopic to each other.(C) Let c : |FL(V ′)| × |FL(V ′′)| → |FL(V ′ ⊕ V ′′)| denote the map produced byα. Denote the action of g ∈ GL(V ′ ⊕ V ′′) on |FL(V ′ ⊕ V ′′)| by |FL(g)|. Thenc and |FL(g)| c are homotopic to each other, if g ∈ Elem(V ′ → V ′ ⊕ V ′′) ⊂GL(V ′ ⊕ V ′′).

Proof. Part (A). By the definition of Hom(K1 ×K2,K3) in section 0, we onlyhave to check that α(σ′ × σ′′) ∪ β(σ′ × σ′′) is a simplex of FL(V ′ ⊕ V ′′) forall simplices σ′ of FL(V ′) and all simplices σ′′ of FL(V ′′). Clearly it sufficesto prove this for maximal simplices, so we assume that both σ′ and σ′′ aremaximal.Note that if we consider any maximal simplex σ′ in FL(V ′), it correspondsto a splitting L′

1, . . . , L′r ∈ SPL(V ′). Similarly any maximal simplex σ′′ of

FL(V ′′) corresponds to a splitting L′′1 , . . . , L

′′s ∈ SPL(V ′′). This determines

the splitting i′(L′1), . . . , i

′(L′r), i

′′(L′′1), . . . , i

′′(L′′s ) of V

′⊕V ′′, giving rise to amaximal simplex τ of FL(V ′ ⊕ V ′′), and clearly α(σ′ × σ′′) and β(σ′ × σ′′) areboth contained in τ . Thus their union is a simplex.(B) follows from (A). We now address (C). We note that c = g c for allg ∈ id + HomA(V

′′, V ′). Denoting by d the map produced by β we see thatd = g d for all g ∈ id +HomA(V

′, V ′′). Because c, d are homotopic to eachother, we see that c and g c are in the same homotopy class when g is in eitherof the two groups above. These groups generate Elem(V ′ → V ′ ⊕ V ′′), and sothis proves (C).

Corollary 9. (i) The group En+1(A) of elementary matrices acts triviallyon the image of the natural map

i∗ : H∗(FL(A⊕n),Z)→ H∗(FL(A

⊕n+1),Z).

(ii) The action of the group E(A) of elementary matrices on H∗(FL(A∞),Z)

is trivial.

Proof. We put V ′ = An and V ′′ = A in the previous lemma. The c in part (B)of the lemma is precisely the i being considered here. By (C) of the lemma,g i is homotopic to i for all g ∈ Elem(An → An+1) = En+1(A). This proves(i). The direct limit of the r-homology of |FL(An)|, taken over all n, is the r-thhomology of |FL(A∞|. Thus (i) implies (ii).

We will find it useful below to have other “equivalent models” of the spacesFL(V ), FL(A∞), by which we mean other simplicial complexes, also definedusing collections of appropriate A-submodules, such that there are natural



homotopy equivalences between the different models of the same homotopytype, compatible with the appropriate group actions, etc.We apply corollary 7 as follows. Let V ∼= An. We put A = SPL(V ), B =FL(V ) and Z = α, F ) : F ∈ [α]. The simplicial complex SZ(B) of corollary 7is FL(V ) by its definition. The simplicial complex SZ(A) is our definition ofSPL(V ). The homotopy equivalence of SPL(V ) and FL(V ) follows from thiscorollary.We define SPL(A∞) to be the collection of sets S satisfying(a) L ∈ S implies that L is a free rank one A-submodule of A∞,(b)⊕L : L ∈ S → A∞ is an isomorphism, and(c) the symmetric difference of S and the standard collection:A(1, 0, 0, ...), A(0, 1, 0, ...), · · · is a finite set.Corollary 7 is then applied to the subset Z ⊂ SPL(A∞)× FL(A∞) consistingof the pairs (S, F ) so that there is a bijection h : S → N so that for everyL ∈ S,L ⊂ Fh(L) and L→ grF

h(L) is an isomorphism.

The above Z defines SPL(A∞). The desired homotopy equivalence of the geo-metric realisations of SPL(A∞) and FL(A∞) comes from the same corollary.We also find it useful to introduce a third model of the homotopy types ofFL(V ) and FL(A∞), the “enriched Tits buildings” ET(V ) and ET(A∞). Thelatter is defined in the last remark of this section.Let V ∼= An as a left A-module. Let E(V ) be the set consisting of ordered pairs

(F, S) = ((0 = F0 ⊂ F1 ⊂ · · · ⊂ Fr = V ), (S1, S2, . . . , Sr)) ,

where F is a partial flag in V , which means that Fi ⊂ V is an A-submodule,such that Fi/Fi−1 is a nonzero free module for each i, and Si ∈ SPL(Fi/Fi−1)is an unordered collection of free A-submodules of Fi/Fi−1 giving rise to a directsum decomposition ⊕L∈Si

L ∼= Fi/Fi−1. Thus S is a collection of splittings ofthe quotients Fi/Fi−1 for each i.We may put a partial order on the set E(V ) in the following way: (F, S) ≤(F ′, T ) if the filtration F is a refinement of F ′, and the data S, T of direct sumdecompositions of quotients are compatible, in the following natural sense —if F ′

i−1 = Fj−1 ⊂ Fj ⊂ · · ·Fl = F ′i , then Ti must be partitioned into subsets,

which map to the sets Sj , Sj+1, . . . , Sl under the appropriate quotient maps.In particular, (F ′, T ) has only finitely many possible predecessors (F, S) in thepartial order.We have a simplicial complex ET (V ) := NE(V ), the nerve of the partiallyordered set E(V ) considered as a category, so that simplices are just nonemptyfinite chains of elements of the vertex (po)set E(V ).Note that maximal elements of E(V ) are naturally identified with elementsof SPL(V ), while minimal elements are naturally identified with elements ofFL(V ). Simplices in FL(V ) are nonempty finite subsets of FL(V ) which have acommon upper bound in E(V ), and similarly simplices in SPL(V ) are nonemptyfinite subsets of SPL(V ) which have a common lower bound in E(V ).



We now show that ET(V ) = BE(V ), the classifying space of the poset E(V ), isanother model of the homotopy type of |FL(V )|.In a similar fashion, we may define a poset E(A∞), and a space ET(A∞), givinganother model of the homotopy type of |FL(A∞)|.We first have a lemma on classifying spaces of certain posets. For any poset(P,≤), and any S ⊂ P , let

L(S) = x ∈ P | x ≤ s ∀ s ∈ S, U(S) = x ∈ P | s ≤ x ∀ s ∈ S

be the upper and lower sets of S in P , respectively. Let Pmin denote thesimplicial complex with vertex set Pmin given by minimal elements of P , andwhere a nonempty finite subset S ⊂ Pmin is a simplex if U(S) 6= ∅. Let |Pmin|denote the geometric realisation of Pmin.

Lemma 10. Let (P,≤) be a poset such that(a) ∀ s ∈ P , the set L(s) is finite(b) if ∅ 6= S ⊂ P with L(S) 6= ∅, then the classifying space BL(S) of L(S) (asa subposet) is contractible.

Then |Pmin| is naturally homotopy equivalent to BP .

Proof. We apply Proposition 1. Take

Λ = L(S)| ∅ 6= S ⊂ P and L(S) 6= ∅.

This is a poset with respect to inclusion. All λ ∈ Λ are finite subsets of P ,so Λ is Artinian. By assumption, the subsets B(λ) ⊂ BP , for λ ∈ Λ, arecontractible. On the other hand, the sets λ ∩ Pmin give simplices in |Pmin|.Thus, both the spaces BP and |Pmin| have Λ-admissible coverings, and arethus homotopy equivalent.

Remark. If a poset P has g.c.d. in the sense that ∅ 6= S ⊂ P and ∅ 6=L(S) implies L(S) = L(t) for some t ∈ P , then condition (b) of the lemma isimmediately satisfied. However E(V ) does not enjoy the latter property.For example, if V = A3 with basis e1, e2, e3, let s = Ae1, Ae2, Ae3 andt = Ae1, A(e1 + e2), Ae3 and let S = s, t ⊂ SPL(V ) ⊂ E(V ). Then L(S)has three minimal elements and two maximal elements. In particular, g.c.d.(s, t) does not exist. In this example, B(L(S)) is an oriented graph in the shapeof the letter M.

Proposition 11. If V is a free A module of finite rank, the poset E(V ) satisfiesthe hypotheses of lemma 10. Thus, |FL(V )| is naturally homotopy equivalentto ET(V ) = B(E(V )).

Proof. Clearly the condition (a) of lemma 10 holds, so it suffices to prove (b).We now make a series of observations.

(i) Regard SPL(V ) as the set of maximal elements of the poset E(V ).We observe that for any s ∈ E(V ), if H(s) = SLP (V ) ∩ U(s), thenwe have that L(s) = L(H(s)). This is easy to see, once one hasunravelled the definitions.



Thus, it suffices to show that for sets S of the type ∅ 6= S ⊂SPL(V ) ⊂ E(V ), we have that B(L(S) is contractible. We assumehenceforth that S ⊂ SPL(V ).

(ii) Given a submoduleW ⊂ V which determines a partial flag 0 ⊂W ⊂ V ,we have a natural inclusion of posets

E(W )× E(V/W ) ⊂ E(V ),

where on the product, we take the partial order

(a1, a2) ≤ (b1, b2)⇔ a1 ≤ b1 ∈ E(W ) and a2 ≤ b2 ∈ E(V/W ).

Note that α ∈ E(V ) lies in the sub-poset E(W )× E(V/W ) preciselywhen W is one of the terms in the partial flag associated to α. Hence,if α lies in the sub-poset, so does the entire set L(α).

(iii) With notation as above, if ∅ 6= S ⊂ SPL(V ) ⊂ E(V ), and L(S) hasnonempty intersection with the image of E(W )×E(V/W ) ⊂ E(V ), thenclearly there exist nonempty subsets S′(W ) ⊂ SPL(W ), S′′(W ) ⊂SPL(V/W ) such that

(4) L(S) ∩ (E(W ) × E(V/W )) = L(S′(W ))× L(S′′(W ))

(iv) If ∅ 6= S ⊂ SPL(V ), then each s ∈ S is a subset of

L(V ) = L ⊂ V |L is a free direct summand of rank 1 of V ,

the set of lines in V . Let

T (S) = ∩s∈Ss = lines common to all members of S,

so that T (S) ⊂ L(V ). Let M(S) denote the direct sum of the elementsof T (S), so that M(S) is a free A-module of finite rank, and 0 ⊂M(S) ⊂ V is a partial flag, in the sense explained earlier; further,T (S) may be regarded also as an element of SPL(M(S)) ⊂ E(M(S)).

(v) We now claim the following: if ∅ 6= S ⊂ SPL(V ) and b ∈ L(S), thenthere exists a unique subset f(b) ⊂ T (S) such that if M(b) is the(direct) sum of the lines in f(b), then

b = (f(b), b′) ∈ SPL(M(b))× E(V/M(b)) ⊂ E(M(b))× E(V/M(b)) ⊂ E(V ).

Indeed, if

b = ((0 = W0 ⊂W1 ⊂ · · · ⊂Wh = V ), (t1, t2, . . . , th)

where ti ∈ SPL(Wi/Wi−1), then since b ∈ L(S), we must have thatt1 ⊂ s for all s ∈ S, which implies that t1 ⊂ T (S). Take M(b) = W1,t1 = f(b) ∈ SPL(M(b)).

Let P(T (S)) be the poset of nonempty subsets of T (S), with respectto inclusion. Then b 7→ f(b) gives an order-preserving map f : L(S)→P(T (S)).

(vi) For any b ∈ L(S), we have

L(b) = L(f(b))× L(b′) ⊂ E(M(b))× E(V/M(b)),

so that if b1 ∈ L(b), then ∅ 6= f(b1) ⊂ f(b).



We will now complete the proof of Proposition 11. We proceed by inductionon the rank of V . Suppose S ⊂ SPL(V ) is nonempty, and L(S) 6= ∅.If M(S) = V , then S = s for some s, and L(S) = L(s) is a cone, hencecontractible. So assume M(S) 6= V .If T ⊂ T (S) is non-empty, and M(T ) ⊂ V the (direct) sum of the lines in T ,then in the notation of (4) above, with W = M(T ), we have S′(W ) = T ,and so f−1(T ) = T × L(S′′(W ))) for some S′′(W ) ⊂ SPL(V/W ).Now by induction, we have that L(S′′(W )) is contractible, provided it is non-empty. Hence the non-empty fiber posets of f are contractible. If ∅ 6= T ⊂T ′ ⊂ T (S), then there is a morphism of posets f−1(T ) → f−1(T ′) given asfollows: if b ∈ f−1(T ), and

b = ((0 = W0 ⊂W1 ⊂ · · · ⊂Wh = V ), (t1, t2, . . . , th)

where ti ∈ SPL(Wi/Wi−1), then since b ∈ f−1(T ), we must have t1 = T ,W1 = M(T ). Now define b′ ∈ f−1(T ′) using the partial flag

0 = W ′ ⊂W1 +M(T ′) ⊂W2 +M(T ′) ⊂ · · · ⊂Wh +M(T ′) = V

and elements t′i ∈ SPL(Wi +M(T ′)/Wi−1 +M(T ′)) induced by the ti. Thisis easily seen to be well-defined, and gives a morphism of posets f−1(T ) →f−1(T ′). In particular, if T ⊂ T ′ ⊂ T (S) and f−1(T ) is non-empty, then so isf−1(T ′).Now take any b ∈ L(S) and put T = f(b), T ′ = T (S) in the above to deducethat f−1(T (S)) 6= ∅. By (iii) above, we see that every f−1X is nonempty (andtherefore contractible as well) for every nonempty X ⊂ T (S).We see that all the fiber posets f−1(T ) considered above are nonempty.This makes f pre-cofibered, in the sense of Quillen (see [19], page 96), withcontractible fibers. Hence by Quillen’s Theorem A, f induces a homotopyequivalence on classifying spaces. But P(T (S)) is contractible (for example,since T (S) is the unique maximal element).

Remark. Proposition 5 and the remarks preceding it apply to the aboveProposition. In particular, we obtain homotopy equivalences f : ET(V ) →FL(V ) so that the induced maps on homology are GL(V )-equivariant.

Remark. We now define the poset E(A∞) and show that ET(A∞) = BE(A∞)is homotopy equivalent to |FL(A∞|.We have already observed that a short exact sequence of free modules of finiterank

0→ V ′ → V → V ′′ → 0

induces a natural inclusion E(V ′) × E(V ′′) → E(V ) of posets. In particular,when V ′′ ∼= A, this yields an inclusion E(V ′) → E(V ).We have ... ⊂ An ⊂ An+1 ⊂ ... ⊂ A∞ as in the definition of FL(A∞). Fromthe above, we obtain a direct system of posets

...E(An) → E(An+1) → ...

and we define E(A∞) to be the direct limit of this system of posets.



We put P = E(A∞) in lemma 10. We note that α ≤ β, α ∈ E(A∞), β ∈ E(An)implies that α ∈ E(An). It follows that P = E(A∞) satisfies the requirementsof the lemma because each E(An) does. It is clear that Pmin = FL(A∞), andfurthermore that Pmin = FL(A∞). This yields the homotopy equivalence of|FL(A∞)| with ET(A∞).By Proposition 5, it follows that ET(A∞)//GL(A) and |FL(A∞)|//GL(A) arealso homotopy equivalent to each other.It has already been remarked that Corollary 7 gives the homotopy equivalenceof |SPL(A∞| with |FL(A∞)|. Combined with Proposition 5, this gives thehomotopy equivalence of |SPL(A∞|//GL(A) with |FL(A∞)|//GL(A). The re-marks preceding that proposition, combined with corollary 9, show that theaction of E(A) on the homology groups of |SPL(A∞)| is trivial.

3. homology of the Borel constrcuction

Let V be a free A-module of rank n. Fix β ∈ SPL(V ) and let N(β) ⊂ GL(V )be the stabiliser of β (when V = An and β is the standard splitting, then N(β)is the subgroup Nn(A) of the last section). That there is a GL(V )-equivariantN(β)-torsor on |FL(V )| has been observed in the previous section. In a similarmanner, one may construct a GL(V )-equivariant N(β)-torsor on ET(V ). Thisgives rise to a N(β)-torsor on ET(V )//GL(V ). Because BN(β) is a classifyingspace for such torsors, we obtain a map ET(V )//GL(V )→ BN(β), well definedup to homotopy.On the other hand, the inclusion of β in ET(V ) gives rise to an inclusionBN(β) = β//N(β) → ET(V )//GL(V ). It is clear that the compositeBN(β) → ET(V )//GL(V ) → BN(β) is homotopic to the identity. ThusBN(β) is a homotopy retract of ET(V )//GL(V ), but not homotopy equivalentto ET(V )//GL(V ). Nevertheless we have the following statement:

Proposition 12. The map BN(β) → ET(V )//GL(V ) induces an isomor-phism on integral homology, provided A is as in theorem 1.

Proof. Fix a basis for V , identifying GL(V ) with GLn(A). Let β ∈ SPL(V )be the element naturally determined by this basis. Regarded as a vertex ofET(V ), let (β, ∗) 7→ β under the natural map

π : ET(V )//GL(V )→ ET(V )/GL(V )

from the homotopy quotient to the geometric quotient, where ∗ ∈ EGL(V ) isthe base point (corresponding to the vertex labelled by the identity element ofGL(V )).For any x ∈ ET(V ), let H(x) ⊂ GL(V ) be the isotropy group of x for theGL(V )-action on ET(V ). Note that since

ET(V )//GL(V ) = (ET(V )× EGL(V )) /GL(V ),

the fiber π−1(π((x, ∗)) may be identified with EGL(V )/H(x), which has thehomotopy type of BH(x).



In particular, the fiber π−1(β) has the homotopy type of BNn(A), Further,the principal Nn(A) bundle on EGL(V )/H(β) is naturally identified with theuniversal Nn(A)-bundle on BNn(A) – its pullback to β×EGL(V ) is the triv-ial Nn(A)-bundle, regarded as an Nn(A)-equivariant principal bundle, whereNn(A) acts on itself (the fiber of the trivial bundle) by translation. This meansthat the composite

π−1(β)→ ET(V )/GL(V )→ BNn(A)

is a homotopy equivalence, which is homotopic to the identity, if we identifyEGL(V )/H(β) with BNn(A).

Thus, the lemma amounts to the assertion that π−1(β) → ET(V )//GL(V )induces an isomorphism in integral homology.Fix α ∈ FL(V ) with α ≤ β in the poset E(V ). Let

P = λ ∈ E(V )|α ≤ λ ≤ β.

One sees easily that (i) BP is contractible, and (ii) the map BP →ET(V )/GL(V ) is a homoemorphism. The first assertion is obvious, since Phas a maximal (as well as a minimal) element, so that BP is a cone. For thesecond assertion, we first note that an element b ∈ P ⊂ E(V ) is uniquely de-termined by the ranks of the modules in the partial flag in V associated to b.Conversely, given any increasing sequence of numbers n1 < . . . < nh = rankV ,there does exist an element of P whose partial flag module ranks are theseintegers. Given any element b ∈ E(V ), there exists an element g ∈ GL(V ) sothat g(b) = b′ ∈ P ; the element b′ is the unique one determined by the sequenceof ranks associated to b. Finally, one observes that if b ∈ P , and g ∈ GL(V )such that g(b) ∈ P , then in fact g(b) = b: this is a consequence of the unique-ness of the element of P with a given sequence of ranks. These observationsimply that BP → ET(V )/GL(V ) is bijective; it is now easy to see that it is ahomeomorphism.We may view ET(V )//GL(V ) as the quotient of BP ×EGL(V ) by the equiv-alence relation

(5) (x, y) ∼ (x′, y′)⇔ x = x′, and y′ = g(y) for some g ∈ H(x).

The earlier map π : ET(V )//GL(V ) → ET(V )/GL(V ) may be viewed now asthe map induced by the projection BP × EGL(V )→ BP . We may, with thisidentification, also identify β with β.Next, we construct a “good” fundamental system of open neighbourhoods ofan arbitrary point x ∈ BP , which we need below. Such a point x lies in therelative interior of a unique simplex σ(x) (called the carrier of x) correspondingto a chain λ0 < λ1 < · · · < λr. Then one sees that the stabiliser H(x) ⊂ GL(V )is given by

H(x) =r⋂

i=0

H(λi),



since any element of GL(V ) which stabilizes the simplex σ(x) must stabilizeeach of the vertices (for example, since the GL(V ) action preserves the partialorder).Let star (x) be the union of the relative interiors of all simplices in BP con-taining σ(x) (this includes the relative interior of σ(x) as well, so it containsx). It is a standard property of simplicial complexes that star (x) is an openneighbourhood of x in BP . Then if z ∈ star (x), clearly σ(z) contains σ(x),and so H(z) ⊂ H(x).Next, for such a point z, and any y ∈ EGL(V ), it makes sense to consider thepath

t 7→ (tz + (1− t)x, y) ∈ σ(z)× EGL(V ) ⊂ BP × EGL(V )

(where we view the expression tz+(1−t)x as a point of σ(z), using the standardbarycentric coordinates). In fact this path is contained in star (x) × y, andgives a continuous map

H(x) : star (x) × EGL(V )× I → star (x)× EGL(V )

which exhibits x × EGL(V ) as a strong deformation retract of star (x) ×EGL(V ). Further, this is compatible with the equivalence relation ∼ in (5)above, so that we obtain a strong deformation retraction

H(x) : π−1(star (x)) × I → π−1(star (x)).

In a similar fashion, we can construct a fundamental sequence of open neigh-bourhoods Un(x) of x in BP , with U1(x) = star (x), and set

Un(x) = H(x)(star (x) × EGL(V )× [0, 1/n)).

The same deformation retraction H determines, by reparametrization, a defor-mation retraction

Hn(x) : π−1(Un(x)) × I → π−1(Un(x))

of π−1(Un(x) onto π−1(x).Thus, if P ′ = P \ β, then

π−1(star (β)) = ET(V )//GL(V ) \ π−1(BP ′),

and from what we have just shown above, the inclusion

π−1(β)→ π−1(star (β)) = ET(V )//GL(V ) \ π−1(BP ′)

is a homotopy equivalence. To simplify notation, we let X = ET(V )//GL(V ),so that we have the map π : X → BP , and X0 = X \ π−1(BP ′). Let π0 =π |X0 : X0 → BP .We are reduced to showing, with this notation, that the inclusion of the (dense)open subset

X0 → X

induces an isomorphism in integral homology. Equivalently, it suffices to showthat this inclusion induces an isomorphism on cohomology with arbitrary con-stant coefficients M . By the Leray spectral sequence, this is a consequence of



showing that the maps of sheaves

Riπ∗MX → Riπ0∗MX0

is an isomorphism, which is clear on stalks x ∈ BP \ BP ′. Now considerstalks at a point x ∈ BP ′. For any point x′ ∈ star (x), note that x lies insome face of σ(x′) (the carrier of x′). We had defined a fundamental system ofneighbourhoods Un(x) of x in BP ; explicitly we have

Un(x) = tx′ + (1 − t)x|0 ≤ t < 1/n and x′ ∈ star (x).

Here, as before, we make sense of the above expression tx′ + (1 − t)x usingbarycentric coordinates in σ(x′).Define

zn(x) =1

2nβ + (1−

1

2n)x.

Note that z ∈ BP \BP ′ = star (β). Further, observe that Un(x)∩BP \BP ′ iscontractible, contains the point z, and for any w ∈ Un(x)∩BP \BP ′, containsthe line segment joining z and w (this makes sense, in terms of barycentriccoordinates of any simplex containing both zn(x) and w; this simplex is eitherthe carrier of w, or the cone over it with vertex β, of which σ(w) is a face).This implies H(w) ⊂ H(zn(x)) = H(x) ∩ H(β), for all w ∈ Un(x). A minormodification of the proof (indicated above) that π−1(x) ⊂ π−1(Un(x)) is astrong deforamtion retract, yields the statement that

π−1(zn(x))→ π−1(Un(x) \BP ′)

is a strong deformation retract. Hence, the desired isomorphism on stalksfollows from:

(6) B(H(x) ∩H(β))→ B(H(x)) induces isomorphisms in integral homology.

We now show how this statement, for the appropriate rings A, is reduced toresults of [13].First, we discuss the structure of the isotropy groups H(x) encountered above.Let λ ∈ P , given by

λ = (F, S) = ((0 = F0 ⊂ F1 ⊂ · · · ⊂ Fr = V ), (S1, S2, . . . , Sr)) ,

where we also have α ≤ λ ≤ β for our chosen elements α ∈ FL(V ) andβ ∈ SPL(V ). We may choose a basis for each of the lines in the splitting β;then α ∈ FL(V ) uniquely determines an order among these basis elements,and thus a basis for the underlying free A-module V , such that the i-th sub-module in the full flag α is the submodule generated by the first i elements inβ. Now the stabilizer H(α) may be viewed as the group of upper triangularmatrices in GLn(A), while H(β) is the group generated by the diagonal sub-group in GLn(A) and the group of permutation matrices, identified with thepermutation group Sn.In these terms, H(λ) has the following structure. The filtration F = (0 = F0 ⊂F1 ⊂ · · · ⊂ Fr = V ) is a sub-filtration of the full flag α, and so determines a“unipotent subgroup” U(λ) of elements fixing the elements of this partial flag,



and acting trivially on the graded quotients Fi/Fi−1. These are represented asmatrices of the form

In1∗ ∗ · · · ∗

0 In2∗ · · · ∗

0 0 In3· · · ∗

.... . .

...0 · · · Inr

where ni = rank (Wi/Wi−1), Iniis the identity matrix of size ni; these are the

matrices which are strictly upper triangular with respect to a certain “ladder”.Next, we may consider the group S(λ) ⊂ Sn of permutation matrices, supportedwithin the corresponding diagonal blocks, of the form

A1 0 0 · · · 00 A2 0 · · · 00 0 A3 · · · 0

.... . .

...0 · · · Ar

where each Aj is a permutation matrix. Finally, we have the diagonal matricesTn(A) ⊂ GLn(A), which are contained in H(λ) for any such λ. In fact H(λ) =U(λ)Tn(A)S(λ), where the group Tn(A)S(λ) normalizes the subgroup U(λ),making H(λ) a semidirect product of U(λ) and Tn(A)S(λ). We also have thatS(λ) normalizes U(λ)Tn(A).In particular, H(α) has trivial associated permutation group S(α) = In,while H(β) has trivial unipotent group U(β) = In associated to it.Now if x ∈ BP , and σ(x) is the simplex associated to the chain λ0 < · · · < λr

in the poset P , then it is easy to see that H(x) is the semidirect product ofU(x) := U(λr) and Tn(A)S(x), with S(x) := S(λ0), since as seen earlier, H(x)is the intersection of the H(λi). In other words, the “unipotent part” and the“permutation group” associated to H(x) are each the smallest possible onesfrom among the corresponding groups attached to the vertices of the carrier ofx. Again we have that S(x) normalizes U(x)Tn(A).We return now to the situation in (6). We see that the groups H(x) =U(x)Tn(A)S(x) and H(x) ∩ H(β) = Tn(A)S(x) both have the same associ-ated permutation group S(x), which normalizes U(x)Tn(A) as well as Tn(A).By comparing the spectral sequences

E2p,q = Hp(S(x), Hq(U(x)Tn(A),Z))⇒ Hp+q(H(x),Z),

E2p,q = Hp(S(x), Hq(Tn(A),Z))⇒ Hp+q(H(x) ∩H(β),Z)

we see that it thus suffices to show that the inclusion

(7) Tn(A) ⊂ U(x)Tn(A)

induces an isomorphism on integral homology.Now lemma 13 below finishes the proof.



To state lemma 13 we use the following notation. Let I = i0 = 0 < i1 < i2 <· · · < ir = n be a subsequence of 0, 1, . . . , n, so that I determines a partialflag

0 ⊂ Ai1 ⊂ Ai2 ⊂ · · · ⊂ Air = An,

where Aj ⊂ An as the submodule generated by the first j basis vectors. LetU(I) be the “unipotent” subgroup of GLn(A) stabilising this flag, and actingtrivially on the associated graded A-module, and let G(I) ⊂ GLn(A) be thesubgroup generated by U(I) and Tn(A) = (A×)n, the subgroup of diagonalmatrices. Then Tn(A) normalises U(I), and G(I) is the semidirect product ofU(I) and Tn(A).

Lemma 13. Let A be a Nesterenko-Suslin ring. For any I as above, the ho-momorphism G(I)→ G(I)/U(I) ∼= Tn(A) induces an isomorphism on integralhomology H∗(G(I),Z)→ H∗(Tn(A),Z).

Proof. We work by induction on n, where there is nothing to prove when n = 1,since we must have G(I) = T1(A) = A× = GL1(A). Next, if n > 1, andI = 0 < n, then U(I) is the trivial group, so there is nothing to prove.Hence we may assume n > 1, r ≥ 2, and thus 0 < i1 < n. There is then anatural homomorphism G(I)→ G(I ′), where I ′ = 0 < i2−i1 < · · · < ir−i1 =n− i1, and G(I ′) ⊂ GLn−i1(A). Let n

′ = n− i1. The induced homomorphismTn(A)→ Tn′(A) is naturally split, with kernel Ti1(A) ⊂ GLi1(A) ⊂ GLn(A).Let

U1(I) = ker (U(I)→ U(I ′)) = ker (G(I)→ GLi1(A)×GLn′(A)) .

Then U1(I) is a normal subgroup of G(I), from the last description, and

G(I)/U1(I) ∼= U(I ′) · Tn(A) = Ti1(A) ×G(I ′).

Now U1(I) may be identified with Mi1,n′(A), the additive group of matricesof size i1 × n′ over A; this matrix group has a natural action of GLi1(A),and thus of the diagonal matrix group Ti1(A), and the resulting semidirectproduct of Ti1(A) with U1(A) is a subgroup of G(I) (in fact, it is the kernel ofG(I)→ G(I ′)). This matrix group Mi1,n′(A) is isomorphic, as Ti1(A)-modules,to the direct sum

⊕i1i=1A

n′

(i),

where An′

(i) is the free A-module of rank n′, with a Ti1(A)-action given bythe the “i-th diagonal entry” character Ti1(A) → A×. Thus, the semidirectproduct Ti1(A)U1(I) has a description as a direct product

Ti1(A)U1(i) ∼= H ×H × · · · ×H = Hni

with H = An′

·A× equal to the naturally defined semidirect product of the freeA module An′

with A×, where A× operates by scalar multiplication.Proposition 1.10 and Remark 1.13 in the paper [13] of Nesterenko and Suslin

implies immediately that H → H/An′ ∼= A× induces an isomorphism on inte-gral homology.We now use the following facts.



(i) If H ⊂ K ⊂ G are groups, with H , K normal in G, and if K → K/Hinduces an isomorphism in integral homology, so does G→ G/H ; thisfollows at once from a comparison of the two spectral sequences

E2r,s = Hr(G/K,Hs(K,Z))⇒ Hr+s(G,Z),

E2r,s = Hr(G/K,Hs(K/H,Z))⇒ Hr+s(G/H,Z).

(ii) If Hi ⊂ Gi are normal subgroups, for i = 1, . . . , n, such thatGi → Gi/Hi induce isomorphisms on integral homology, then forG =

∏ni=1 Gi, H =

∏ni=1 Hi, the map G → G/H induces an isomor-

phism on integral homology. This follows from the Kunneth formula.

The fact (ii) implies that Ti1(A)U1(I) → Ti1(A) induces an isomorphismon integral homology. Then (i) implies that G(I) → Ti1(A) × G(I ′) in-duces an isomorphism on integral homology. By induction, we have thatG(I ′) → G(I ′)/U(I ′) induces an isomorphism on integral homology. HenceTi1(A) × G(I ′) → Ti1 × G(I ′)/U(I ′) also induces an isomorphism on integralhomology. Thus, we have shown that the composition G(I) → G(I)/U(I) =Tn(A) induces an isomorphism on integral homology.

4. SPL(A∞)+ and the groups Ln(A)

We first note that there is a small variation of Quillen’s plus construction.Let (X, x) be a pointed CW complex, (X0, x) a contractible pointed subcom-plex, G a group of homeomorphisms of X which acts transitively on the pathcomponents of X , and let H be a perfect subgroup of G, such that H stabilizesX0.Then X//G is clearly path connected, and comes equipped with(i) a natural map θ : X//G → BG = EG/G, induced by the projectionX × EG→ EG(ii) a map (X0 × EG)/H → X//G, induced by the H-stable contractible setX0 ⊂ X(iii) a homotopy equivalence BH → (X0 ×EG)/H , such that the composition

BH → X//Gθ→ BG is homotopic to the natural map BH → BG

(iv) a natural map (X, x) → (X//G, x0) determined by the base point of EG.Note that, in particular, there is a natural inclusion H → π1(X//G, x0), whichgives a section over H ⊂ G of the surjection θ∗ : π1(X//G, x0)→ π1(BG, ∗) =G.

Lemma 14. In the above situation, there is a pointed CW complex (Y, y), to-gether with a map f : (X//G, x0)→ (Y, y) such that(i) the natural composite map

H → π1(X//G, x0)f→ π1(Y, y)

is trivial(ii) if g : (X//G, x0)→ (Z, z) such that H is in the kernel of

π1(X//G, x0)→ π1(Z, z)



then g factors through f , uniquely upto a pointed homotopy(iii) f induces isomorphisms on integral homology; more generally, if L is anylocal system on Y , the map on homology with coefficients H∗(X//G, f∗L) →H∗(Y, L) is an isomorphism(iv) h : (X, x) → (X ′, x′) is a pointed map of such CW complexes with G-actions, such that h is G-equivariant, then there is a map (Y, y) → (Y ′, y′),making (X, x) 7→ (Y, y) is functorial (on the category of pointed CW complexeswith suitable G actions, and equivariant maps), and f yields a natural trans-formation of functors.

The pair (Y, y) is obtained by applying Quillen’s plus construction to

(X//G, x0) with respect to the perfect normal subgroup H of π1(X//G, x0)which is generated by H . Part (ii) of the lemma is in fact the universal prop-erty of the plus construction. As is well-known, this may be done in a functorialway. We sometimes write (Y, y) = (X//G, x0)

+ to denote the above relation-ship.In what follows, the pair (G,H) is invariably (GLn(A), An) for 5 ≤ n ≤ ∞.Here An is the alternating group contained in Nn(A). The normal subgroupsof GLn(A) generated by An and En(A) coincide with each other. It followsthat if we take X = X0 to be a point, the Y given by the above lemma is justthe “original” BGLn(A)

+.Recall that there is a natural action of GL(A) on the simplicial complexSPL(A∞), and hence on its geometric realization |SPL(A∞)|. We applylemma 14 with G = GL(A), H = A∞ the infinite alternating group, X =|SPL(A∞)|, and X0 = x0 is the vertex of X fixed by N(A) and obtain thepointed space

(Y (A), y) = (|SPL(A∞)|//G), x0)+.

Taking X ′ to be a singleton in (iii) of the above lemma, we get a canonical map

ϕ : (Y (A), y)→ (BGL(A)+, ∗)

of pointed spaces.Let (SPL(A∞)+, z) denote the homotopy fibre of ϕ. We define

Ln(A) = πn(SPL(A∞)+, z) ∀ n ≥ 0.

The homotopy sequence of the fibration SPL(A∞)+ → Y (A) → BGL(A)+

combined with the path-connectedness of Y (A) yields:

Corollary 15. There is an exact sequence

· · · → Kn+1(A)→ Ln(A)→ πn(Y (A), y)→ Kn(A) · · ·

· · · → L1(A)→ π1(Y (A), y)→ K1(A)→ L0(A)→ 0

where L0(A) is regarded as a pointed set.

Lemma 16. The natural map |SPL(A∞)| → SPL(A∞)+ induces an isomor-phism on integral homology.



Proof. We may identify the universal covering of BGL(A)+ with BE(A)+,where BE(A)+ is the plus construction (see lemma 14) applied to BE(A) withrespect to the infinite alternating group (or, what is the same thing, with

respect to E(A) itself). Let ϕ : Y → BE(A)+ be the corresponding pullbackmap obtained from ϕ.We first note that SPL(A∞)+ is also naturally identified with the homotopy

fiber of ϕ. There is then a homotopy pullback ϕ : Y → BE(A) of ϕ withrespect to BE(A) → BE(A)+. Thus, our map SPL(A∞) → SPL(A∞)+ maybe viewed as the natural map on fibers associated to a map

(8) SPL(A∞)//E(A)→ Y

of Serre fibrations over BE(A).From a Leray-Serre spectral sequence argument, we see that since (fromlemma 14) BE(A) → BE(A)+ induces a isomorphism on integral homology,

so does Y → Y . Since also SPL(A∞)//E(A) → Y is a homology isomor-

phism (from lemma 14 again), we see that SPL(A∞)//E(A) → Y induces anisomorphism on integral homology.Now we use that the map (8) is a map between two total spaces of Serrefibrations over a common base, inducing a homology isomorphism on these totalspaces. We also know that the monodromy representation of π1(BE(A)) =

E(A) on the homology of the fibers is trivial, in both cases: for Y this isbecause it is a pullback from a Serre fibration over a simply connected base,while for SPL(A∞), this is one of the key properties we have already established(see the finishing sentence of section 2). The proof is now complete modulothe remark below, which is a straightforward consequence of the Leray-Serrespectral sequence of a fibration.

Remark. Let p : E → B and p′ : E′ → B be fibrations with fibers F andF ′ respectively over the base-point b ∈ B. Let v : E → E′ be a map so thatp′ v = p. Assume that B is path-connected. Then E → E′ is a homologyisomorphism implies F → F ′ is a homology isomorphism under the followingadditional assumption:M 6= 0 implies H0(π1(B, b),M) 6= 0 for every π1(B, b)-subquotient M ofHi(F ), Hj(F

′) for all i, j.

5. The H-space structure

Recall that BGL(A)+ has an H-space structure in a standard way, obtainedfrom the direct sum operation on free modules of finte rank; this was con-structed in [6].The aim of this section is to prove the proposition below.

Proposition 17. The space Y (A) has an H-space structure, such that Y (A)→BGL(A)+ is homotopic to an H-map, for the standard H-space structure onBGL(A)+.



We first remark that if V is a free A-module of finite rank, then|SPL(V )|//GL(V ) is homeomorphic to the classifying space of the followingcategory SPL(V ): its objects are simplices in SPL(V ) (thus, certain finitenonempty subsets of SPL(V )), and morphisms σ → τ are defined to be ele-ments g ∈ GL(V ) such that g(σ) ⊂ τ , that is, such that g(σ) is a face of thesimplex τ of SPL(V ).Let Aut(V ) be the category with a single object ∗, with morphisms given by el-ements of GL(V ), so that the classifying space BAut(V ) is the standard modelfor BGL(V ). There is a functor FV : SPL(V ) → Aut(V ), mapping everyobeject σ to ∗, and mappng an arrow σ → τ in SPL(V ) to the correspondingelement g ∈ GLV (). The fiber F−1

V (∗) is the poset of simplices of SPL(V ),whose classifying space is thus homeomorphic to |SPL(V )|.It is fairly straightforward to verify that BSPL(V ) is homeomorphic to|SPL(V )|//GL(V ) (where we have used the classifying space of the transla-tion category of GL(V ) as the model for the contractible space E(GL(V ))).

One way to think of this is to consider the category SPL(V ), whose objectsare pairs (σ, h) with σ a simplex of SPL(V ), and h ∈ GL(V ), with a uniquemorphism (σ, h) → (τ, g) precisely when g−1h(σ) ⊂ τ . It is clear that by con-sidering the full subcategories of objects of the form (σ, g), where g ∈ GL(V )is a fixed element, each of which is naturally equivalent to the poset of sim-

plices in SPL(V ), that the classifying space of SPL(V ) is homeomorphic to|SPL(V )|×E(GL(V ))). Now it is a simple matter to see (e.g., use the criterion

of Quillen, given in [19], lemma 6.1, page 89) that BSPL(V ) → BSPL(V ),given by (σ, h) 7→ h−1(σ), is a covering space which is a principal GL(V )-bundle, where the deck transformations are given by the natural action ofGL(V ) on |SPL(V )| × E(GL(V )).L(V ) denotes the collection of A-submodules L ⊂ V so that L is free of rankone and V/L is a free module. Now we note that if V ′, V ′′ are free A-modulesof finite rank, we note that there is a natural inclusion L(V ′) ⊔ L(V ′′) →L(V ′ ⊕ V ′′). This in turn yields a natural map

ϕV ′,V ′′ : SPL(V ′)× SPL(V ′′)→ SPL(V ′ ⊕ V ′′),

given by ϕV ′,V ′′(s, t) = s ⊔ t.It follows easily from the definition of SPL that the above map on verticesinduces a simplicial map

ΦV ′,V ′′ : SPL(V ′)× SPL(V ′′)→ SPL(V ′ ⊕ V ′′).

As explained in section 0, at the level of geometric realisations, this has twodescriptions. The first description may be used to show that the counterpartof lemma 8(C) is valid for SPL, namely the homotopy class of the inclusion

|SPL(V ′)| × |SPL(V ′′)| → |SPL(V ′ ⊕ V ′′)|

remains unaffected by composition with the action of g ∈ Elem(V ′ → V ′⊕V ′′)on |SPL(V ′ ⊕ V ′′)|.The second description however is more useful in this context. Let us abbreviatenotation and denote the (partially ordered) set of simplices of SPL(V ) simply



by S(V ). The desired map S(V ′) × S(V ′′) → S(V ′ ⊕ V ′′) is given simply by(σ, τ) 7→ ϕV ′,V ′′(σ× τ). The resulting map B(S(V ′)×S(V ′′))→ BS(V ′⊕V ′′)is the second description of

|SPL(V ′)| × |SPL(V ′′)| → |SPL(V ′ ⊕ V ′′)|

for (a) BC′ × BC′′ ∼= B(C′ × C′′) and (b) BS(V ) is simply the barycentricsubdivision of |SPL(V )|.This latter description also allows us to go a step further and define the func-tor SPL(V ′) × SPL(V ′′) → SPL(V ′ ⊕ V ′′), given on objects by (σ, τ) 7→ϕV ′,V ′′(σ × τ) as before; on morphisms, it is given by the natural mapGL (V ′)×GL (V ′′)→ GL(V ′ ⊕ V ′′). Hence on classifying spaces, it induces aproduct

|SPL(V ′)|//GL(V ′)×|SPL(V ′′)|//GL(V ′′)→ |SPL(V ′⊕V ′′)|//GL(V ′⊕V ′′)|.

This is clearly compatible with the product

BGL(V ′)×BGL(V ′′)→ BGL(V ′ ⊕ V ′′)

under the natural maps induced by the functors SPL → Aut for the three freemodules.One verifies that SPL(A) =

∐V SPL(V ), with respect to the bifunctor

+ : SPL(A)× SPL(A)→ SPL(A)

induced by direct sums on free modules, and the functors ΦV ′,V ′′ , form a sym-metric monoidal category.An equivalent category, also denoted SPL(A) by abuse of notation, is thatwhose objects are pairs (V, σ), where V is a free A-module of finite rank, andσ ∈ SPL(V ) a simplex, and where morphisms (V, σ)→ (W, τ) are isomorphismsf : V →W of A-modules such that f(σ) is a face of τ .For the purposes of stabilization, we slightly modify the above to consider therelated maps

ϕm,n : SPL(Am)× SPL(An)→ SPL(A∞)

given by mapping the basis vector ei ∈ Am in the first factor to the basis vectore2i−1 ∈ A∞, for each 1 ≤ i ≤ m, and the basis vector ej ∈ An in the secondfactor to the basis vector e2j ∈ A∞. A pair of splittings of Am, An determineone for the free module spanned by the images of the two sets of basis vectors;now one extends this to a splitting of A∞ by adjoining the remaining basisvectors of A∞ (that is, adjoining those vectors not in the span of the earlierimages). If our first two splittings are those given by the basis vectors, whichcorrespond to the base points in |SPL(Am)| and |SPL(An)|, the resulting pointin SPL(A∞) is again the base point of |SPL(A)|.The corresponding functors

Φm,n : SPL(Am)× SPL(An)→ SPL(A∞)

are compatible with similar functors

Aut(Am)×Aut(An)→ Aut(A∞)



which, on classifying spaces, yield the diagram of product maps, preservingbase points,

|SPL(Am)//GLn(A)| × |SPL(An)|//GLn(A) → |SPL(A∞|//GL(A)↓ ↓

BGLm(A) ×BGLn(A) → BGL(A)

where the bottom arrow is the one used in [6] to define the H-space structureon BGL(A)+.As we increase m, n, the corresponding diagrams are compatible with respectto the obvious stabilization maps |SPL(Am)| → |SPL(Am+1)|, |SPL(An) →|SPL(An+1)|. Hence we obtain on the direct limits a diagram

|SPL(A∞)//GL(A)| × |SPL(A∞)|//GL(A) → |SPL(A∞|//GL(A)↓ ↓

BGL(A)×BGL(A) → BGL(A)

From lemma 14, it follows that there is an induced diagram at the level of plusconstructions

Y (A)× Y (A) → Y (A)↓ ↓

BGL(A)+ ×BGL(A)+ → BGL(A)+

Here we have, as remarked above, taken the homotopy equivalent model|SPL(A∞|//GL(A) for the homotopy type earlier denoted Y (A). We abusenotation and use the same symbol to denote this model as well.It is shown in [6] that the botton arrow defines an H-space structure onBGL(A). We claim that, by analogous arguments, the top arrow also definesan H-space structure on Y (A). Granting this, the map Y (A) → BGL(A)+ isthen an H-map between path connected H-spaces, and so the homotopy fiberZ(A) has the homotopy type of an H-group as well (and this was what we setout to prove here).To show that the product Y (A)× Y (A)→ Y (A) defines an H-space structure,we need to show that left or right translation on Y (A) (with respect to thisproduct) by the base point is homotopic to the identity. This is also the mainpoint in [6], for the case of BGL(A)+. We first show:

Lemma 18. An arbitrary inclusion j : 1, 2, . . . , n → N determines an in-clusion of A-modules An → A∞, given on basis vectors by ei 7→ ej(i), whichinduces a map

|SPL(An)|//GLn(A)→ Y (A)

which is homotopic (preserving the base point) to the map induced by standardinclusion in : An → A∞.

Proof. We can find an automorphism g of A∞ contained in the infinite alter-nating group A∞, such that g j = in, where g acts on A∞ by permuting thebasis vectors (note that the induced self-map of |SPL(A∞)| × EGL(A) fixesthe base point). Regarding g as an element of π1(|SPL(A∞)|//GL(A)), this



implies that the maps (in)∗ and j∗, considered as elements of the set of pointedhomotopy classes of maps

[|SPL(An)|//GLn(A), |SPL(A∞)|//GL(A)] ,

are related by g∗(j∗) = (in)∗, where g∗ denotes the action of the fundamentalgroup of the target on the set of pointed homotopy classes of maps. However,g is in the kernel of the map on fundamental groups associated to the map

|SPL(A∞)|//GL(A)→ (|SPL(A∞|//GL(A))+.

Hence the induced maps

|SPL(An)|//GL(A)→ (|SPL(A∞)|//GL(A))+

determined by in and j are homotopic.

Corollary 19. The map Y (A)→ Y (A) defined by an arbitrary injective mapα : N → N is homotopic, preserving the base point, to the identity.

Proof. We first note that if for n ≥ 5, we let Yn(A) = (|SPL(An)|//GLn(A))+

be the result of applying lemma 14 to |SPL(An)|//GLn(A) for the alternatinggroup An, then there are natural maps Yn(A)→ Y (A), preserving base points,and inducing an isomorphism lim

−→n

π∗(Yn(A)) = π∗(Y (A)).

We claim that if αn : 1, 2 . . . , n → N is the inclusion induced by restrictingα, then the induced map (αn)∗ : Yn(A) → Y (A) is homotopic, preserving thebase points, to the natural map Yn(A) → Y (A). This follows from lemma 18,combined with the defining universal property of the plus construction, givenin lemma 14.This implies that the map α : Y (A) → Y (A) must then induce isomorphismson homotopy groups, and hence is a homotopy equivalence, by Whitehead’stheorem.Thus, we have a map from the set of such injective maps α to the group ofbase-point preserving homotopy classes of self-maps of Y (A). This is in fact ahomomorphism of monoids, where the operation on the injective self-maps ofN is given by composition of maps.Now we use a trick from [6]: any homomorphism of monoids from the monoidof injective self-maps of N to a group is a trivial homomorphism, mapping allelements of the monoid to the identity. This is left to the reader to verify (orsee [6]).

We note that the above monoidal category SPL(A) can be used to give another,perhaps more insightful construction of the homotopy type Y (A), analogous toQuillen’s S−1S construction for BGL(A)+. We sketch the argument below.We first take SPL0(A) to be the full subcategory of SPL(A) consisting ofpairs (V, σ) where σ ∈ SPL(V ), i.e.,σ is a 0-simplex in SPL(V ). This fullsubcategory is in fact a monoidal subcategory, which is a groupoid (all ar-row are isomorphisms). Also, SPL(A) is a symmetric monoidal category, inthat the sum operation is commutative upto coherent natural isomorphisms.



Then, using Quillen’s results (see Chapter 7 in [19], particularly Theorem 7.2),one can see that SPL0(A)

−1SPL(A) is a monoidal category whose classify-ing space is a connected H-space, which is naturally homology equivalent to|SPL(A∞)|//GL(A). This then forces this classifying space to be homotopyequivalent to Y (A), such that the H-space operations are compatible upto ho-motopy. This is analogous to the identification made in Theorem 7.4 in [19]of S−1S with K0(R) × BGL(R)+ for a ring R, and appropriate S. (We donot get the factor K0 appearing in our situation since we work only with freemodules).

6. Theorem 1 and the groups Hn(A×)

Proof of Theorem 1. In view of Proposition 17, we see that SPL(A∞)+, thehomotopy fiber of the H-map Y (A)→ BGL(A)+, is a H-space as well. It followsthat L0(A) = π0(SPL(A

∞)+) is a monoid. Furthermore, the arrow K1(A) →L0(A) in Corollary 15 is a monoid homomorphism. Thus this corollary producesan exact sequence of Abelian groups.SPL(An) has a canonical base point fixed under the action of Nn(A). As insections 3 and 4, this gives a natural inclusion BNn(A)→ SPLn(A)//GLn(A).This is a homology isomorphism by lemma 12. Taking direct limits over alln ∈ N, we see that BN(A)→ SPL(A∞)//GL(A) is a homology isomorphism.Applying Quillen’s plus construction with respect to the normal subgroup ofN(A) generated by the infinite alternating group, we obtain a space BN(A)+.ThatBN(A)+ has a canonical H-space structure follows easily by the method ofthe previous section. Now the map BN(A)+ → Y (A) obtained by lemma 14(ii)is a homology isomorphism of simple path-connected CW complexes and istherefore a homotopy equivalence (see [4]) Theorem 4.37, page 371 and The-orem 4.5, page 346). This gives the isomorphism Hn(A

×) → Ln(A). Thetheorem now follows from corollary 15.We now turn to the description of the groups. Let X = B(A×). Let X+ =X ⊔ ∗ be the pointed space with ∗ as its base-point. Let QX+ be the directlimit of ΩnΣnX+ where Σ denotes reduced suspension.

Proposition 20. Hn(A×) ∼= πn(QX+).

This statement was suggested to us by Proposition 3.6 of [17].A complete proof of the proposition was shown us by Peter May. A condensedversion of what we learnt from him is given below.Theorem 2.2, page 67 of [8] asserts that α∞ : C∞X+ → QX+ is a groupcompletion. This is proved in pages 50-59, [10]. The C∞ here is a particularcase of the construction 2.4, page 13 of [9], given for any operad. For C∞(Y ),where Y is a pointed space, the easiest definition to work with is found inMay’s review of [16]. It runs as follows. Let V = ∪∞n=0R

n. Let Ck(Y ) be thecollection of ordered pairs (c, f) where c ⊂ V has cardinality k and f : c→ Yis any function. We identify (c, f) with (c′, f ′) if(i) c′ ⊂ c, (ii) f |c′ = f ′, and (iii) f(a) = ∗ for all a ∈ c, a /∈ c′. Here ∗ stands forthe base-point of Y . Then C∞Y is the space obtained from the disjoint union



of the Ck(Y ), k ≥ 0 by performing these identifications. This is a H-space.In our case, when Y = X ⊔ ∗, it is clear that C∞Y is the disjoint union ofall the Ck(X) as a topological space. Thanks to “infinite codimension” onegets easily the homotopy equivalence of Ck(X) with Xk//Sk where Sk is thepermutation group of 1, 2, .., k. Now assume that X is any path-connectedspace equipped with a nondegenerate base-point x ∈ X . This x gives aninclusion of Xn → Xn+1. Denote by X∞ the direct limit of the Xn. Thus X∞

is a pointed space equipped with the action of the infinite permutation groupS∞ = ∪nSn. Put Z = X∞//S∞. As in section 4, we obtain Z+ by the use ofthe infinite alternating group. As in section 5, we see that this is a H-space. Itis an easy matter to check that the group completion of ⊔kCkX is homotopyequivalent to Z× Z+. This shows that πn(QX+) ∼= πn(Z

+) for all n > 0.The proposition is the particular case: X = B(A×).

7. Polyhedral structure of the enriched Tits building

From what has been shown so far, we see that it is of interest to determinethe stable rational homology of the flag complexes |FL(An)| (or equivalently,of |SPL(An)|, or ET(An)). We will construct a spectral sequence that, inprinciple, gives an inductive procedure to do so.But first we introduce some notation and a definition for posets.Let P be a poset. For p ∈ P , we put e(p) = BL(p) where L(p) = q ∈ P |q ≤ pand ∂e(p) = BL′(p) where L′(p) = L(p) \ p. If ∂e(p) is homeomorphic to asphere for every p ∈ P , we say the poset P is polyhedral. We denote by d(p)the dimension of e(p). When P is polyhedral, the space BP gets the structureof a CW complex with e(p) : p ∈ P as the closed cells. Its r-skeleton is BPr

where Pr = p ∈ P : d(p) ≤ r. The homology of BP is then computed by theassociated complex of cellular chains Cell•(BP ), where

Cellr(BP ) =⊕

p| dim e(p)=r

Hr(e(p), ∂e(p),Z).

Lemma 21. E(An) is a polyhedral poset in the above sense. Its dimension isn− 1.

Proof. First consider the case when p ∈ SPL(An) is a maximal element inE(An). Then p is an unordered collection of n lines in An (here, as in §2, a“line” denotes a free A-submodule of rank 1 which is a direct summand, andthe set of lines in An is denoted by L(An)). Note that the subset p ⊂ L(An) ofcardinality n determines a poset p, whose elements are chains q• = q1 ⊂ q2 ⊂· · · ⊂ qr = p of nonempty subsets, where r• ≤ q• if each qi is an rj for some j,i.e., the “filtration” r• “refines” q•. We claim that, from the definition of thepartial order on E(An), the poset p is naturally isomorphic to the poset L(p).Indeed, an element q ∈ E(An) consists of a pair, consisting of a partial flag

0 = W0 ⊂W1 ⊂ · · · ⊂Wr = An



such that Wi/Wi−1 is free, and a sequence t1, . . . , tr with ti ∈ SPL(Wi/Wi−1).The condition that this element of E(An) lies in L(p) is that each Wi is adirect sum of a subset of the lines in p, say qi ⊂ p, giving the chain of subsetsq1 ⊂ q2... ⊂ qr = p; the splitting ti is uniquely determined by the lines inqi \ qi−1.Let ∆(p) be the (n− 1)-simplex with p as its set of vertices. Now the chains ofnon-empty subsets of p correspond to simplices in the barycentric subdivisonsd∆(p), where the barycentre b corresponds to the chain p. Hence, from thedefinition of p, it is clear that it is isomorphic to the poset whose elements aresimplices in the barycentric subdivision of ∆n with b as a vertex, with partialorder given by reverse inclusion. Hence Bp is naturally identified with thesubcomplex of the second barycentric subdivision sd2∆(p) which is the unionof all simplices containing the barycentre. This explicit description implies inparticular that BL′(p) is homeomorphic to Sn−2 (with a specific triangulation).Before proceeding to the general case, we set up the relevant notation fororientations. For a set q of cardinality r, we put det(q) = ∧rZ[q] where Z[q]denotes the free Abelian group with q as basis. we observe that there is anatural isomorphism:

Hn−1(e(p), ∂e(p)) ∼= Hn−1(∆(p), ∂∆(p)) = det(p).

Now let p ∈ E(An) be arbitrary, corresponding to a partial flag

0 = W0 ⊂W1 ⊂ · · · ⊂Wr = An.

and splittings ti ∈ SPL(Wi/Wi−1). Then the natural mapr∏

i=1

E(Wi/Wi−1)→ E(An)

is an embedding of posets, where the product has the ordering given by(a1, . . . , ar) ≤ (b1, . . . , br) precisely when ai ≤ bi in E(Wi/Wi−1) for each i.One sees that, by the definition of the partial order in E(An), the induced map

r∏

i=1

L(ti)→ L(p)

is bijective. Hence there is a homeomorphism of pairs

(BL(p), BL′(p)) =

r∏

i=1

(BL(ti), BL′(ti)),

and so BL′(p) ∼= Sn−r−1, and BL(p) is an n− r-cell.

We now proceed to construct the desired spectral sequence. We use the fol-lowing notation: if p ∈ E(V ), where V is a free A-module of finite rank, andW1 ⊂ V is the smallest non-zero submodule in the partial flag associated top, define t(p) = rankW1 − 1. Clearly t : E(V ) → Z is monotonic. HenceFrE(V ) = p ∈ E(V )|t(p) ≤ r is a sub-poset. Define

FrET(V ) = BFrE(V ) = ∪e(p)|t(p) ≤ r,



so that

F0ET(V ) ⊂ F1ET(V ) ⊂ · · ·Fn−1ET(V ) = ET(V )

is an increasing finite filtration of the CW complex ET(V ) by subcomplexes.Hence there is an associated spectral sequence

E1r,s = Hr+s(FrET(V ), Fr−1ET(V ),Z)⇒ Hr+s(ET(V ),Z).

Our objective now is to recognise the above E1 terms.It is convenient to use the complexes of cellular chains for these sub CW-complexes, which are thus sub-chain complexes of Cell•(ET(V )). For simplicityof notation, we write Cell•(V ) for Cell•(ET(V )). We have the description

E1r,s = Hr+s(gr

Fr Cell•(V )).

We will now exhibit grFr Cell•(V ) as a direct sum of complexes. Let W ⊂ Vbe a submodule such that W,V/W are both free, and rankW = r + 1. Letq ∈ SPL(W ). The we have an inclusion of chain complexes

Cell•(e(q)) ⊗ Cell•(V/W ) ⊂ Cell•(W )⊗ Cell•(V/W ) ⊂ FrCell•(V ).

It is clear that

imageCell•(∂e(q))⊗ Cell•(V/W ) ⊂ Fr−1Cell•(V ),

so that we have an induced homomorphism of complexes

(Cell•(e(q))/Cell•(∂e(q)))⊗ Cell•(V/W )→ grFr Cell•(V ).

Composing with the natural chain homomorphism

Hr(e(q), ∂e(q),Z)[r] → (Cell•(e(q))/Cell•(∂e(q)))

for each q, we finally obtain a chain map

I :⊕

(W,q∈SPL(W ))

Hr(e(q), ∂e(q),Z)[r] ⊗ Cell•(V/W )→ grFr Cell•(V ).

Finally, it is fairly straightforward to verify that I is an isomorphism of com-plexes.We deduce that the E1 terms have the following description:

E1r,s =

⊕

rankW = r + 1q ∈ SPL(W )

det(q)⊗Hs(Cell•(V/W ),Z).

We define Lr(V ) to be the collection of q ⊂ L(V ) of cardinality (r + 1) forwhich (a) and (b) below hold:(a) ⊕L : L ∈ q → V is injective. Its image will be denoted by W (q)(b) V/W (q) is free of rank (n− r − 1).Summarising the above, we obtain:



Theorem 2. There is a spectral sequence with E1 terms

E1r,s =

⊕

q ∈ Lr(V )

det(q) ⊗Hs(ET(V/W (q)),Z).

that converges to Hr+s(ET(V )). We note that E1r,s = 0 whenever (r + s) ≥

(n− 1) with one exception: (r, s) = (n− 1, 0). Here V ∼= An.

8. Compatible homotopy

It is true 3 that i : ET(W )×ET(V/W ) → ET(V ) has the property that g i isfreely homotopic (not preserving base points) to i whenever g ∈ Elem(W → V ).There are several closed subsets of ET(An) with the property that homotopyclass of the inclusion morphism into ET(An) remains unaffected by compositionwith the action of g ∈ En(A). To prove that the union of a finite collection ofsuch closed subsets has the same property, one would require the homotopiesprovided for any two members of the collection to agree on their intersection.This is the problem we are concerned with in this section.We proceed to set up the notation for the problem.With q ∈ Lr(V ) as in theorem 2, we shall define the subspaces U(q) ⊂ ET(V )as follows. Let W (q) = ⊕L|L ∈ q. We regard q as an element of SPL(W (q))and thus obtain the cell e(q) = BL(q) ⊂ ET(W (q)). This gives the inclusionET′(q) = e(q)× ET(V/W (q)) ⊂ ET(W (q))× ET(V/W (q)) ⊂ ET(V ).We put U(q) = ∪ET′(t)|∅ 6= t ⊂ q.

Main Question: Let i : U(q) → ET(V ) denote the inclusion. Is it truethat g i is homotopic to i for every g ∈ Elem(V, q)?We focus on the apparently weaker question below.Compatible Homotopy Question: Let M ⊂ V be a submodule complementaryto W (q). Let g′ ∈ GL((W (q)) be elementary, i.e. g′ ∈ Elem(W (q), q). Defineg ∈ GL(V ) by gm = m for all m ∈ M and gw = g′w for all w ∈ W (q). Is ittrue that g i is homotopic to i?Assume that the second question has an affirmative answer in all cases. Inparticular, this holds when M = 0. Here V = W (q) and g = g′ is an arbitraryelement of Elem(V, q). Let t be a non-empty subset of q. Then U(t) ⊂ U(q). Wededuce that j : U(t) → ET(V ) is homotopic to g j for all g ∈ Elem(V, q). ButElem(V, t) = Elem(V, q). Thus the Main Question has an affirmative answerfor (q, i) replaced by (t, j), which of course, up to a change of notation, coversthe general case.

Proposition 22. The compatible homotopy question has an affirmative answerif q ∈ Lr(V ) and r ≤ 2.

The rest of this section is devoted to the proof of this proposition. To proceed,we will require to introduce the class C.

3this only requires the analogue of lemma 8(A) for the enriched Tits building. Moregeneral statements are contained in lemmas 23 and 24 .



This is our set-up. Let X be a finite set, let Vx be a finitely generated freemodule for each x ∈ X and let V = ⊕Vx : x ∈ X.Let s = Πx∈Xs(x) ∈ Πx∈XSPL(Vx). For each x ∈ X , we regard s(x) asa subset of L(V ) and put Fs = ∪s(x)|x ∈ X. Thus Fs ∈ SPL(V ).The collection of maps f : Πx∈XET(Vx) → ET(V ) with the property thatf(Πx∈XBL(s(x))) ⊂ BL(Fs) for all s ∈ Πx∈XSPL(Vx) is denoted by C. Seelemma 10 and proposition 11 and its proof for relevant notation.Every maximal chain C of subsets of X (equivalently every total ordering ofX) gives a member i(C) ∈ C. For instance, if X = 1, 2, ..., n and C consistsof the sets 1, 2, ..., k for 1 ≤ k ≤ n, we putDk = ⊕k

i=1Vi and E = Πni=1ET(Di/Di−1), denote by

u : Πx∈XET(Vx) → E and v : E → ET(V ) the natural isomorphism andnatural inclusion respectively, and put i(C) = v u.

Lemma 23. The above space C is contractible.

Proof. The aim is to realise C as the space of Λ-compatible maps for a suitableΛ and appeal to Proposition 1. Let Λ(x) = L(S)|∅ 6= S ⊂ SPL(Vx), ∅ 6=L(S). Proposition 11 asserts that the subspaces Bλ(x) : λ(x) ∈ Λ(x) givean admissible cover of ET(Vx).For λ = Πx∈Xλ(x) ∈ Λ = Πx∈XΛ(x), we put I(λ) = Πx∈XBλ(x) and deducethat I(λ) : λ ∈ Λ gives an admissible cover of Πx∈XET(Vx).We define next a closed J(λ) ⊂ ET(V ) for every λ ∈ Λ with the properties:(A): J(λ) ⊂ J(µ) whenever λ ≤ µ and(B): J(λ) is contractible for every λ ∈ Λ.For each λ(x) ∈ Λ(x), let Uλ(x) be its set of upper bounds in SPL(Vx). Itfollows that that LUλ(x)) = λ(x). As observed before, we haveF : Πx∈XSPL(Vx)→ SPL(V ). Thus given λ = Πxλ(x) ∈ Λ,we set H(λ) = F (Πx∈XUλ(x)) ⊂ SPL(V ) and then put J(λ) = BLH(λ) ⊂ET(V ).The space of Λ-compatible maps Πx∈XET(Vx) → ET(V ) is seen to coincidewith C. That the J(λ) satisfy property (A) stated above is straightforward.The contractibility of J(λ) for all λ ∈ Λ is guaranteed by proposition 11 onceit is checked that these sets are nonempty. But we have already noted that Cis nonempty. Let f ∈ C. Now I(λ) 6= ∅ and f(I(λ)) ⊂ J(λ) implies J(λ) 6= ∅.Thus the J(λ) are contractible, and as said earlier, an application of Proposi-tion 1 completes the proof of the lemma.

We remark that the class C of maps Πni=1ET(Wi) → ET(⊕n

i=1Wi) has beendefined in general.We will continue to employ the notation: V = ⊕Vx : x ∈ X all through thissection. Let P be a partition of X . Each p ∈ P is a subset of X and we put

Vp = ⊕Vx|x ∈ p and ET(P ) = ΠET(Vp)|p ∈ P.

When Q ≤ P is a partition of X (i.e. Q is finer than P ), we shall define thecontractible collection C(Q,P ) of maps f : ET(Q)→ ET(P ) by demanding (a)



that f is the product of maps f(p)

f(p) : ΠET(Vq) : q ⊂ p and q ∈ Q → ET(Vp)

and also (b) each f(p) is in the class C. For this one should note that Vp =⊕Vq : q ∈ Q and q ⊂ p.We observe next that there is a distinguished collection D(Q,P ) ⊂ C(Q,P ).To see this, recall that we had the embedding i(C) for every maximal chain Cof subsets of X (alternatively, for every total ordering of X). Given Q ≤ P ,denote the set of total orderings of q ∈ Q : q ⊂ p by T (p), for every p ∈ P .The earlier C 7→ i(C) now yields, after taking a product over p ∈ P ,i : ΠT (p) : p ∈ P → C(Q,P ), and we denote by D(Q,P ) ⊂ C(Q,P ) theimage of i.The lemma below is immediate from the definitions.

Lemma 24. Given partitions R ≤ Q ≤ P of X, if f is in C(R,Q) (resp. inD(R,Q))and g is in C(Q,P ) (resp. in D(Q,P )), then it follows that g f is inC(R,P ) (resp. D(R,P )).

We will soon have to focus on the fixed points of certain unipotent g ∈ GL(V )on ET(V ). For instance, if x, y ∈ X and x 6= y, we may consider g = idV + hwhere h(V ) ⊂ Vy and h(Vz) = 0 for all z ∈ X, z 6= x. Let C be a chain of subsetsof X , so that X ∈ C. This chain C gives rise to a partition P (C) of X andalso i(C) ∈ D(P (C), X) in a natural manner. Let Cx = ∩S ∈ C : x ∈ S.Then Cx ∈ C because C is a chain. Define Cy in a similar manner. We saythe chain C is (x, y)-compatible if Cy ⊂ Cx and Cx 6= Cy . This condition onC ensures that the embedding i(C) : ET(P (C))→ ET(V ) has its image withinthe fixed points of the above g ∈ GL(V ).Now let Q be a partition of X so that q ∈ Q, x ∈ q implies y /∈ q. Weshall define next the class of (x, y)-compatible C maps ET(Q) → ET(V ) inthe following manner. Let Λ be the set of chains C of subsets of X so thatX ∈ C and Q ≤ P (C) (i.e. Q is finer than the partition P (C)). For eachC ∈ Λ, let Z(C) be the collection of i(C) f where f ∈ C(Q,P (C)). Finally,let Z = ∪Z(C) : C ∈ Λ. This set Z is defined to be the collection of (x, y)-compatible maps of class C from ET(Q) to ET(V ). Every z ∈ Z is a mapz : ET(Q)→ ET(V ) whose image is contained in the fixed points of the aboveg on ET(V ). Furthermore, in view of lemma 24, this collection of maps iscontained in C(Q, X).

Lemma 25. Let Q be a partition of X that separates x and y. Then the collec-tion of (x, y)-compatible class C maps ET(Q)→ ET(V ) is contractible.

Proof. In view of the fact that each C(Q,P ) is contractible, by cor 3, it followsthat the space of (x, y)-compatible chains is homotopy equivalent to BΛ, whereΛ is the poset of chains C in the previous paragraph. It remains to show thatBΛ is contractible.We first consider the case where Q is the set of all singletons of X . Let S bethe collection of subsets S ⊂ X so that y ∈ S and x /∈ S. For S ∈ S, let



F(S) be the collection of chains C of subsets of X so that S ∈ C and X ∈ C.We see that Λ is precisely the union of F(S) taken over all S ∈ S. Let D bea finite subset of S. We see that the intersection of the BF(S), taken overS ∈ D, is nonempty if and only if D is a chain. Furthermore, when D is achain, this intersection is clearly a cone, and therefore contractible. By cor 3,we see that BΛ has the same homotopy type as the classifying space of theposet of chains of S. But this is simply the barycentric subdivision of BS. Butthe latter is a cone as well, with y as vertex. This completes the proof thatBΛ is contractible, when Q is the finest possible partition of X .We now come to the general case, when Q is an arbitrary partition of X thatseparates x, y. So we have x′, y′ ∈ Q with x ∈ x′, y ∈ y′ and x′ 6= y′. The setΛ is identified with the collection of chains C′ of subsets of Q so that(a) Q ∈ C′, and (b) there is some L ∈ C′ so that x′ /∈ L and y′ ∈ L.Thus the general case follows from the case considered first: one replaces(X, x, y) by (Q, x′, y′).

In a similar manner, we may define, for every ordered r-tuple (x1, x2, ..., xr)of distinct elements of X , the set of (x1, x2, ..., xr)-compatible chains C–wedemand that for each 0 < i < r, there is a member S of the chain so that xi /∈ Sand xi+1 ∈ S. Let Q be a partition of X that separates x1, x2, ..., xr. Thenthe poset of chains C , compatible with respect to this ordered r-tuple, and forwhich Q ≤ i(P ), is also contractible. One may see this through an inductiveversion of the proof of the above lemma. A corollary is that the collection of(x1, ..., xr)-compatible class C maps ET(Q) → ET(V ) is also contractible. Weskip the proof. This result is employed in the proof of Proposition 22 for r = 2(which has already been verified in the above lemma), and for r = 3, with#(Q) ≤ 4. Here it is a simple verification that the poset of chains that arisesas above has its classifying space homeomorphic to a point or a closed interval.

We are now ready to address the proposition. For this purpose, we as-sume that there is c ∈ X so that Vx

∼= A for all x ∈ X \ c. To obtainconsistency with the notation of the proposition, we set q = X \ c. Theclosed subset U(q) ⊂ ET(V ) in the proposition is the union of ET′(t) takenover all ∅ 6= t ⊂ q. For such t, we have W (t) = V (t) = ⊕Vx : x ∈ t. Recallthat ET′(t) is the product of the cell e(t) ⊂ ET(V (t)) with ET(V/V (t)). Toproceed, it will be necessary to give a contractible class of maps D → ET(V )for certain closed subsets D ⊂ U(q).The closed subsets D ⊂ U(q) we consider have the following shape. For each∅ 6= t ⊂ q, we first select a closed subset D(t) ⊂ e(t) and then take D to be theunion of the D(t) × ET(V/V (t)), taken over all such t. This D remains unaf-fected if D(t) is replaced by its saturation sD(t). Here sD(t) is the collectionof a ∈ e(t) for which a × ET(V/V (t)) is contained in D.When ∅ 6= t ⊂ q, we denote by p(t) the partition of X consisting of all thesingletons contained in t, and in addition, the complement X \ t. Then thereis a canonical identification j(t) : ET(p(t))→ ET(V/V (t)).



A map f : D → ET(V ) is said to be in class C if for every ∅ 6= t ⊂ q and forevery a ∈ sD(t), the map ET(p(t))→ ET(V ) given by b 7→ f(a, j(t)b) belongsto C(p(t), X). By lemma 24, we see that it suffices to impose this conditionon all a ∈ D(t), rather than all a ∈ sD(t).We observe that for every a ∈ e(t), the map ET(p(t)) → ET(V ) given byb 7→ (a, j(t)b) belongs to C(p(t), X). As a consequence, we see that theinclusion D → ET(V ) is of class C.When concerned with (x, y)-compatible maps, we will assume that D(t) = ∅whenever t and x, y are disjoint. Under this assumption, a map f : D →ET(V ) is said to be (x, y)-compatible of class C if ET(p(t)) → ET(V ) givenby b 7→ f(a, j(t)b) is a (x, y)-compatible map in C(p(t), X) for all pairs (a, t)such that a ∈ sD(t).In a similar manner, we define (x, y, z)-compatible maps of class C as well. Forthis, it is necessary to assume that D(t) is empty whenever the partition p(t)does not separate (x, y, z), equivalently if x, y, z \ t has at least two elements.

Lemma 26. Assume furthermore that D(t) is a simplicial subcomplex of e(t).Then the space of maps D → ET(V ) in class C is contractible. The same is trueof the space of such maps that are (x, y)-compatible, or (x, y, z)-compatible.

Proof. We denote by d the cardinality of t : D(t) 6= ∅. We proceed byinduction on d, beginning with d = 0 where the space of maps is just onepoint.We choose t0 of maximum cardinality so that D(t0) 6= ∅. Let D′ be the unionof D(t) × ET(V/V (t)) taken over all t 6= t0. Let C(D′) and C(D) denote thespace of class C maps D′ → ET(V ) and D → ET(V ) respectively. By theinduction hypothesis, C(D′) is contractible. We observe that the intersectionof D′ and e(t0)×ET(V/V (t0)) has the form G×ET(V/V (t0) where G ⊂ e(t0)is a subcomplex. Furthermore, G∪D(t0) is the saturated set sD(t0) describedearlier.For a closed subset H ⊂ e(t0), denote the space of C-maps H×ET(V/V (t0))→ET(V ) by A(H). Note that A(H) = Maps(H, C(p(t0), X)). By lemma 23, thespace C(p(t0), X) is itself contractible. It follows that A(H) is contractible.In particular, both A(G) and A(sD(t0)) are contractible. The natural mapA(sD(t0)) → A(G) is a fibration, because the inclusion G → sD(t0) is acofibration. The fibers of A(sD(t0)) → A(G) are thus contractible. It followsthat

A(sD(t0))×A(G) C(D′)→ C(D′)

which is simply C(D)→ C(D′), enjoys the same properties: it is also a fibrationwith contractible fibers. Because C(D′) is contractible, we deduce that C(D) isitself contractible. This completes the proof of the first assertion of the lemma.The remaining assertions follow in exactly the same manner by appealing tolemma 25.

Proof of Proposition 22. Choose x 6= y with x, y ∈ q. Let g = idV + α whereα(V ) ⊂ Vy and α(Vk) = 0 for all k 6= x ∈ X . To prove the proposition, it



suffices to show that g i is homotopic to i where i : U(q) → ET(V ) is thegiven inclusion. This notation x, y, α, g will remain fixed throughout the proof.

Case 1. Here q = x, y. Now x, y are separated by the partitions p(t)for every non-empty t ⊂ q. By the second assertion of the above lemma, thereexists f : U(q)→ ET(V ) of class C and (x, y)-compatible. The given inclusioni : U(q)→ ET(V ) is also of class C. By the first assertion of the same lemma,f is homotopic to i. Now the image of f is contained in the fixed-points of gand so we get g f = f . It follows that g i is homotopic to i. This completesthe proof of the proposition when 1 = r = #(q)− 1.

Case 2. Here q = x, y, z with x, y, z all distinct.We take Y1 to be the union of e(t) × ET(V/V (t)) taken over all t ⊂ q, t 6=z, t 6= ∅. We put Y2 = ET(Vz)× ET(V/Vz) and Y3 = Y1 ∩ Y2. We note thatU(q) = Y1 ∪ Y2.The given inclusion i : U(q) → ET(V ) restricts to ik : Yk → ET(V ) for k =1, 2, 3. The required homotopy is a path γ : I → Maps(U(q),ET(V )) so thatγ(0) = i and γ(1) = gi. Equivalently we require paths γk in Maps(Yk,ET(V ))for k = 1, 2 so that(a) γk(0) = ik and γk(1) = g ik for k = 1, 2 and(b) both γ1 and γ2 restrict to the same path in Maps(Y3,ET(V )).In view of the fact that Y3 → Y1 is a cofibration, the weaker conditions (a′)and (b′) on fundamental groupoids suffice for the existence of such a γ:(a′): γk ∈ π1(Maps(Yk,ET(V )); ik, g ik) for k = 1, 2(b′): both γ1 and γ2 restrict to the same element of π1((Maps(Y3,ET(V )); i3, gi3)We have the spaces: Zk = Maps(Yk,ET(V )) for k = 1, 2, 3.These spaces comeequipped with the data below:(A)The GL(V )-action on ET(V ) induces a GL(V )-action on Zk

(B)The maps of class C give contractible subspaces Ck ⊂ Zk for k = 1, 2, 3.(C) We have ik ∈ Ck for k = 1, 2, 3.(D) The natural maps Zk → Z3 for k = 1, 2 are GL(V )-equivariant, they takeik to i3 and restrict to maps Ck → C3.Note that the GL(V )-action on Zk turns the disjoint union:Gk = ⊔π1(Zk; ik, hik)|h ∈ GL(V ) into a group: given ordered pairs (hj , vj) ∈Gk, i.e. hj ∈ GL(V ) and vj ∈ π1(Zk; ik, hjik) for j = 1, 2, we get h1v2 ∈π1(Zk;h1ik, h1h2ik) and obtain thereby v = (h1v2).v1 ∈ π1(Zk; ik, h1h2ik) andthis produces the required binary operation (h1, v1) ∗ (h2, v2) = (h1h2, v).The projection Gk → GL(V ) is a group homomorphism. The following elemen-tary remark will be used in an essential manner when checking condition (b′).The data (H,F,∆) where(i) H ⊂ GL(V ) is a subgroup,(ii)F ∈ Zk is a fixed-point of H , and(iii)∆ ∈ π1(Zk;F, ik)produces the lift H → Gk of the inclusion H → GL(V ) by h 7→ (h, (h∆).∆−1).



Finally we observe that there are natural homomorphisms Gk → G3 induced byZk → Z3 for k = 1, 2.Construction of γ1.The partitions p(t) for t 6= z separate x, y. By lemma 26, we have a (x, y)-compatible class C-map f : Y1 → ET(V ). Both i1 and f belong to C1 andthus we get δ ∈ π1(C1; f, i1). Now f is fixed by our g ∈ GL(V ), so we also getgδ ∈ π1(gC1; f, gi1). The path (gδ).δ−1 is the desired γ1 ∈ π1(Z1; i1, gi1).Construction of γ2.Recall that g = idV + α. We choose m : Vx → Vz and n : Vz → Vy sothat nm(a) = α(a) for all a ∈ Vx. We extend m,n by zero to nilpotentendomorphisms of V , once again denoted by m,n : V → V and put u =idV + n, v = idV +m and note that g = uvu−1v−1.Note that the partition p(z) separates both the pairs (x, z) and (z, y).We thus obtain f ′, f ′′ ∈ C2 so that f ′ is (x, z)-compatible and f ′′ is (y, z)-compatible and also δ′ ∈ π1(C2; f ′, i2) and δ′′ ∈ π1(C2; f ′′, i2). Noting thatf ′, f ′′ are fixed by v, u respectively, we obtainǫ′ = (vδ′).δ′−1 ∈ π1(Z2; i2, vi2) andǫ′′ = (uδ′′).δ′′−1 ∈ π1(Z2; i2, ui2).Thus v′ = (v, ǫ′) and u′ = (u, ǫ′′) both belong to G2. We obtain γ2 byu′ ∗ v′ ∗ u′−1 ∗ v′−1 = (g, γ2) ∈ G2Checking the validity of (b′).Let γ13, γ23 ∈ π1(Z3; i3, gi3) be the images of γ1 and γ2 respectively. We haveto show that γ13 = γ23.Consider the spacesH,H′,H′′ consisting of ordered pairs (f3, δ3), (f ′

3, δ′

3), (f′′

3 , δ′′

3 )

respectively, where f3, f′3, f

′′3 are all in C3,

f3 is (x, y)-compatible, f ′3 is (x, z)-compatible, and f ′′

3 is (y, z)-compatible, andδ3, δ

′3, δ

′′3 are all paths in C3 that originate at f3, f

′3, f

′′3 respectively, and they

all terminate at i3. By lemma 26, we see that the spaces H,H′,H′′ are allcontractible.For t = x, z, y, z, x, y, z, the partition p(t) separates (x, y, z). Note thatY3 is contained in the union of these three ET′(t). By lemma 26, there is a(x, y, z)-compatible F ∈ C3. Let ∆ be a path in C3 that originates at F andterminates at i3. We see that (F,∆) ∈ H ∩H′ ∩H′′.Note that H → π1(Z3; i3, gi3) given by (f3, δ3) 7→ (gδ3).δ

−13 is a constant map

becauseH is contractible. The (f, δ) employed in the construction of γ1 restrictsto an element of H. Also, (F,∆) belongs to H. It follows that γ13 = (g∆).∆−1.In a similar manner, we deduce that if ǫ′3, ǫ

′′3 denote the images of ǫ′, ǫ′′ in the

fundamental groupoid of Z3, thenǫ′3 = (v∆).∆−1 ∈ π1(Z3; i3, vi3) and ǫ′′3 = (u∆).∆−1 ∈ π1(Z3; i3, ui3)Thus G2 → G3 takes v′, u′ ∈ G2 to (v, (v∆).∆−1), (u, (u∆).∆−1) ∈ G3 respec-tively. It follows that their commutator [u′, v′] maps to (g, γ23) ∈ G3 under thishomomorphism.We apply the remark preceeding the construction of γ1 to the subgroup H gen-erated by u, v and F and ∆ as above. We conclude that γ23 equals (g∆).∆−1.



That the latter equals γ13 has already been shown. Thus γ13 = γ23 and thiscompletes the proof of the Proposition.

9. Low dimensional stabilisation of homology

This section contains applications of corollary 9, proposition 22 and Theorem 2to obtain some mild information on the homology groups of ET(V ). The no-tation L(V ),Lr(V ),W (q), det(q) introduced to state Theorem 2 will be freelyused throughout. The spectral sequence in theorem 2 with coefficients in anAbelian group M will be denoted by SS(V ;M). When V = An, this is furtherabbreviated to SS(n;M), or even to SS(n) when it is clear from the contextwhat M is.The concept of a commutative ring with many units is due to Van der Kallen.An exposition of the definition and consequences of this term is given in[12]. We note that this class of rings includes semilocal rings with infiniteresidue fields. The three consequences of this hypothesis on A are listed asI,II,III below. These statements are followed by some elementary deductions.Throughout this section, we will assume that our ring A has this property.

I: SLn(A) = En(A).This permits a better formulation of Lemma 8 in many instances.Ia: Let 0 → W → P → Q → 0 be an exact sequence of free A-moduleswith of ranks a, a+ b, b. Let d = g.c.d.(a, b). The group H of automorphismsof this exact sequence that induce homotheties on both W and Q may beregarded as a subgroup of GL(P ). This group acts trivially on the image ofthe embedding i : ET(W ) × ET(Q) → ET(P ). Furthermore det(g)|g ∈ Hequals (A×)d. Thus, if a, b are relatively prime, by lemma 8, we see that g iis freely homotopic to i for all g ∈ GL(V ).We shall take rank(W ) = 1 in what follows. Here ET(W )×ET(Q) is canonicallyidentified with ET(Q). The induced ET(Q) → ET(P ) gives rise on homologyto an arrow Hm(ET(Q))→ Hm(ET(P )) which has a factoring:

Hm(ET(Q)) ։ H0(PGL(Q), Hm(ET(Q)))→

→ H0(PGL(P ), Hm(ET(P )) → Hm(ET(P )).

The kernel of Hm(ET(Q)) → Hm(ET(P )) does not depend on the choice ofthe exact sequence. Denoting this kernel by KHm(Q) ⊂ Hm(ET(Q)) thereforegives rise to unambiguous notation. We abbreviate Hm(ET(An)),KHm(An)to Hm(n),KHm(n) respectively.Ib: In the spectral sequence SS(n), we have:

(1) H0(PGLn−1(A), Hm(n− 1)) ∼= H0(PGLn(A), E10,m).

(2) E∞0,m is the image of Hm(n− 1)→ Hm(n).

(3) E10,m → E∞

0,m factors as follows:

E10,m ։ E2

0,m ։ H0(PGLn(A), E10,m) ։ E∞

0,m.

(4) If H0(PGLn(A), Epp,m+1−p) = 0 for all p ≥ 2, then

the given arrow H0(PGLn(A), E10,m)→ E∞

0,m is an isomorphism.



(5) Assume that Hm(n − 2) → Hm(n − 1) is surjective. Then the arrowE2

0,m → H0(PGLn(A), E10,m) in (3) above is an isomorphism.

The factoring in part (3) above is a consequence of the factoring ofHm(ET(Q))→ Hm(ET(P )) in part I(a).For part (4), one notes that the compositeE2

2,m−1 → E20,m → H0(PGLn(A), E

10,m) vanishes because

H0(PGLn(A), E22,m−1) itself vanishes. Thus we obtain a factoring:

E20,m → E3

0,m → H0(PGLn(A), E10,m). Proceeding inductively, we obtain

the factoring:E2

0,m → E∞0,m → H0(PGLn(A), E

10,m). In view of (3), we see that part (4)

follows.For part (5), it suffices to note that for every L0, L1 ∈ L(An) with n > 1, thereis some L2 ∈ L(An) with the property that both L0, L2 and L1, L2 belongto L1(An). This fact is contained in consequence III of many units.

II: A is a Nesterenko-Suslin ring.Let r > 0, p ≥ 0. Put N = (p + 1)! and n = r + p + 1. Let Fr be thecategory with free rank A-modules of rank r as objects; the morphisms in Fr

are A-module isomorphisms. Let F be a functor from Fr to the category ofZ[ 1

N] -modules. Assume that F (a.idD) = idFD for every a ∈ A× and for every

object D of Fr. In other words, the natural action of GLr(A) on F (Ar) factorsthrough the action of PGLr(A).For a free A-module V of rank n, define Ind′F (V ) by

Ind′F (V ) = ⊕det(q)⊗ F (V/W (q)) : q ∈ Lp(V ).

An alternative description of Ind′F (V ) is as follows. Fix some q ∈ Lp(V ). LetG(q) be the stabiliser of q in GL(V ). Then det(q) and F (V/W (q)) are G(q)-modules in a natural manner. We have a natural isomorphism of Z[GL(V )]-modules:

Ind′F (V ) ∼= Z[GL(V )]⊗Z[G(q)] [det(q)⊗Z F (V/W (q))].

IIa: If Hi(PGLr(A), F (Ar)) = 0 for all i < m, thenHi(PGLn(A), Ind

′F (An)) = 0 for all i < p+m. Furthermore,

Hp+m(PGLn(A), Ind′F (An)) ∼= Hm(PGLr(A), F (Ar))⊗ Symp(A×))

By Shapiro’s lemma, the result of [13] cited earlier, and the fact that the grouphomology Hi(M,C) is isomorphic to C ⊗ Λi(M) for all commutative groupsM and Z[1/i!]-modules C given trivial M -action, IIa reduces to the statement:Let Σ(q) denote the group of permutations of a set q of (p+ 1) elements. LetM be an Abelian group on which (p+ 1)! acts invertibly. ThenH0(Σ(q), det(q)⊗Λi(M q)) vanishes when i < p and is isomorphic to Symp(M)when i = p.

III: The standard application of the many units hypothesis, see ( [20] forinstance) is that general position is available in the precise sense given below.Let V ∼= An. We denote by K(V ) the simplicial complex whose set of vertices



is L(V ). A subset S ⊂ L(V ) of cardinality (r + 1) is an r-simplex of K(V ) ifevery T ⊂ S of cardinality t+ 1 ≤ n belongs to Lt(V ). If L ⊂ K(V ) is a finitesimplicial subcomplex, then there is some e ∈ L(V ) with the property thats ∪ e is a (r + 1)-simplex of K(V ) for every r-simplex s of L. This gives anembedding Cone(L) → K(V ) with e as the vertex of the cone. Thus K(V ) iscontractible. The complex of oriented chains of this simplicial complex will be

denoted by C•(V ). Thus the reduced homologies Hi(C•(V )) vanish for all i.The group D(V ) = Zn−1C•(V ) = Bn−1C•(V ) comes up frequently.IIIa:

(1) Let p < n. Let M be a Z[1/N ]-module where N = (p + 1)!.Then Hj(PGL(V ), Cp ⊗M) vanishes for j < p and is isomorphic toSymp(A×)⊗M when j = p.

(2) Hj(PGL(V ), Cn ⊗M) vanishes for all j ≥ 0 and for all Z[1/(n+ 1)!]-modules M .

(3) H0(PGL(V ), D(V ) ⊗M) for any Abelian group M is isomorphic toM/2M if n is even, and vanishes if when n is odd

(4) H0(PGL(V ), BpC•(V ) ⊗M) = 0 for every Z[1/2]-module M and forevery 0 ≤ p < n.

(5) H1(PGL(V ), ZpC•(V ) ⊗M) = 0 for every Z[1/(p + 2)!]- module Mand 1 ≤ p ≤ n− 2.

Note that (1) above follows from IIa when F is the constant functor M .For (2), one observes that PGL(V ) acts transitively on the set of n-simplices ofthe simplicial complex K(V ). The stabiliser of an n-simplex is the permutationgroup Σ on (n+ 1) letters. The claim now follows from Shapiro’s lemma.The presentation Cp+2(V ) ⊗M → Cp+1(V ) ⊗M → BpC•(V ) ⊗M → 0 andthe observation H0(PGL(V ), Cp+1(V )⊗M) ∼= M/2M whenever p < n sufficeto take care of (3) and (4).For assertion (5), one applies the long exact sequence of group homology to theshort exact sequence:0→ Zp+1C•(V )⊗M → Cp+1(V )⊗M → ZpC•(V )⊗M → 0.One therefore obtains the exact sequence:H1(PGL(V ), Cp+1(V ) ⊗ M) → H1(PGL(V ), ZpC•(V ) ⊗ M) →H0(PGL(V ), Zp+1C•(V ) ⊗ M). The end terms here vanish by (1) and(4).IIIb:

(1) ET(V ) is connected.(2) E2

0,0 = Z, E2n−1,0 = D(V ), E2

m,0 = 0 if m 6= 0, n − 1 for the spectralsequence SS(V ).

(3) H1(ET(V )) ∼= Z/2Z if rank(V ) > 2.

Note that (1) is a consequence of (2). Part (2) is deduced by induction onrank(V ) = n. The induction hypothesis enables the identification of the E1

m,0

terms of the spectral sequence for V (together with differentials) with theCm(V ) (together with boundary operators) when m < n. Thus (2) follows.



For part (3), consider the spectral sequence SS(3). Here E22,0 = D(A3) and

H0(PGL3(A), D(A3)) = 0 by IIIa(3). Thus the hypothesis of Ib(4) holds forSS(3). Consequently,H1(3) ∼= E∞

1,0∼= H0(PGL2(A), H1(2)) = H0(PGL2(A), D(A2)) ∼= Z/2Z,

the last isomorphism given by IIIa(3) once again. The isomorphism H1(n) ∼=Z/2Z for n > 3 is contained in the lemma below for N = 1.

Lemma 27. Let M be an Abelian group. Let N ∈ N. For 0 < r < N , we aregiven m(r) ≥ 0 so that Hr(d;M)→ Hr(d+ 1;M)is a surjection if d = r +m(r) + 1 , andan isomorphism if d > r +m(r) + 1.Let m(N) = max0,m(1) + 1,m(2) + 1, ...,m(N − 1) + 1.Then HN (d;M)→ HN (d+ 1;M) is(a) an isomorphism if d > N +m(N) + 1,(b) is a surjection if d = N +m(N) + 1.(c) The surjection in (b) above factors through an isomorphismH0(PGLd(A), HN (d))→ HN (d+ 1;M) if if M is a Z[1/2]-module.

Proof. Consider the spectral sequence SS(V ;M) that computes the homologyof ET(V ) with coefficients in M . Here V is free of rank N + h + 2, whereh ≥ m(N). We make the following claim:Claim:If E2

s,r 6= 0 and 0 < s and r < N , then s + r ≥ N + 1 + h − m(N).

Furthermore, when equality holds, H0(PGL(V ), E2s,r ⊗ Z[1/2]) = 0.

We assume the claim and prove the lemma. We take h = m(N). All the E2s,r

with s + r = N are zero except possibly for (s, r) = (0, N). Part (b) of thelemma now follows from Ib(2). We consider next the E2

s,r with s+ r = N + 1and s ≥ 2 (or equivalently with r < N). It follows that Es

s,r is a quotient of

E2s,r. The second assertion of the claim now show that H0(PGL(V ), Es

s,r) = 0if M is a Z[1/2]-module. Part (c) of the lemma now follows from Ib(4).We take h > m(N) and prove part (a) by induction on h. The inductivehypothesis implies that HN (N + h;M)→ HN (N + h+ 1;M) is surjective. ByIb(5), it follows that HN (N +1+h;M)→ E2

0,N is an isomorphism. Now there

are no nonzero E2s,r with s + r = N + 1 and s ≥ 2. Thus E2

0,N = E∞0,N . It

follows that HN (N + 1 + h;M)→ HN (N + 2 + h;M) is an isomorphism.It only remains to prove the claim. We address this matter now.For r = 0, both assertions of the claim are valid by IIIb(2)and IIIa(3).So assume now that 0 < r < N . Let SH(r) = Hr(d;M) for d = r +m(r) + 2.In view of our hypothesis, the chain complexE1

0,r ← E11,r ← ...← E1

p−1,r ← E1p,r

for N + h+ 1 = p+ r +m(r) + 1 is identified withC0(V ) ⊗ SH(r) ← ... ← Cp−1(V ) ⊗ SH(r) ← ⊕det(q) ⊗Hr(ET(V/W (q));M)|q ∈ Lp(V ).As in IIIb(2), it follows that E2

s,r = 0 whenever 0 < s < p. Furthermore, we

deduce the following exact sequence for E2p,r:



⊕det(q) ⊗ KHr(V/W (q);M)|q ∈ Lp → E2p,r → ZpC• ⊗ SH(r) → 0. By

IIIa(4), we see that H0(PGL(V ), E2p,r) = 0 if M is a Z[1/2]-module.

Note that p + r = N + h −m(r) ≥ N + 1 + h −M(N). This completes theproof of the claim, and therefore, the proof of the lemma as well.

The Proposition below is an application of Proposition 22. The notation hereis that of Theorem 2. We regard Br

p,q and Zrp,q as subgroups of E1

p,q for allr > 1. The notation KHm(Q) has been introduced in Ia, the first applicationof many units.

Proposition 28. Let rank(V ) = n. Let M be a Z[1/2]-module. In the spectralsequence SS(V ;M), we have:

(1) ⊕det(q)⊗KHm(V/W (q))|q ∈ L1(V ) ⊂ B∞1,m if n > 1.

(2) E∞1,m = 0 if n > 2 and M is a Z[1/6]-module.

(3) If, in addition, it is assumed thatHm+1(n− 2;M)→ Hm+1(n− 1;M) is surjective, then⊕det(q)⊗KHm(V/W (q))|q ∈ L2(V ) ⊂ B∞

2,m.

Proof. Let q ∈ Lr(V ). We have U(q) ⊂ ET(V ) as in Proposition 22. Thespectral sequence of Theorem 2 was constructed from an increasing filtration ofsubspaces of ET(V ). Intersecting this filtration with U(q) we obtain a spectralsequence that computes the homology of U(q). Its terms will be denoted byEa

b,c(q). One notes that E1b,m(q) is the direct sum of det(u)⊗Hm(ET(V/W (u)))

taken over all u ⊂ q of cardinality (b+ 1).We denote the terms of the spectral sequence in theorem 2 by Ea

b,c(V ). The

given data also provides a homomorphism Eab,c(q) → Ea

b,c(V ) of E1-spectral

sequences. We assume that M is a Z[1/(r + 1)!]-module.We choose a basis e1, e2, ..., en of V so thatq = Aei : 1 ≤ i ≤ r + 1. Let G ⊂ GL(V ) be the subgroup of g ∈ GL(V ) sothat(A) g(q) = q, (B)g(ei) = ei for all i > r + 1, (C), the matrix entries of g are0, 1,−1 and (D) det(g) = 1. Now G acts on the pair U(q) ⊂ ET(V ). Thusthe above homomorphism of spectral sequences is one such in the category ofG-modules.We observe:(a) G is a group of order 2(r + 1)!(b) there are no nonzero G-invariants in E1

i,m(q) for i > 0, and consequentlythe same holds for all G-subquotients, in particular for Ea

i,m(q) for all a > 0 aswell.Proof of part 1. Take r = 1. Proposition 22 implies that the image ofHm(U(q)) → Hm(ET(V )) has trivial G-action. In view of (b) above, thisshows that E∞

1,m(q) → E∞1,m(V ) is zero. But E∞

1,m(q) = Z∞1,m(q) = det(q) ⊗

KHm(V/W (q)). It follows that det(q)⊗KHm(V/W (q)) ⊂ B∞1,m(V ). Part (1)

follows.Proof of part (2). We take r = 2. Here we haveZ∞1,m(q) = Z2

1,m(q). Appealing to Proposition 22 and observation (b) once



again, we see that the image of the homomorphism Z21,m(q)→ Z2

1,m(V ) is con-tained in B∞

1,m. Part (2) therefore follows from the claim below.

Claim: ⊕Z21,m(q)|q ∈ L2(V ) → Z2

1,m(V ) is surjective.Denote the image ofHm(n−2;M)→ Hm(n−1;M) by I. A simple computationproduces the exact sequences:0→ ⊕det(u)⊗KHm(V/W (u) : u ∈ L1(V ), u ⊂ q → Z2

1,m(q)→ det(q)⊗I →0, and0→ ⊕det(u)⊗KHm(V/W (u) : u ∈ L1(V ) → Z2

1,m(V )→ Z1C•(V )⊗ I → 0.

The claim now follows from the above description of Z21,m(q) and Z2

1,m(V ).Thus part (2) is proved.Proof of part (3).We take r = 2 once again. The surjectivity of Hm+1(n −2;M) → Hm+1(n − 1;M) implies that E2

0,m+1(q) has trivial G-action. By

observation (b), we see that d22,m : E22,m(q) → E2

0,m+1(q) is zero. It follows

that E∞2,m(q) = Z2

2,m(q) here. Proposition 22 and observation (b) once again

show that the image of Z22,m(q) → Z2

2,m(V ) is contained in B∞2,m(V ). Because

Z22,m(q) = det(q)⊗KHm(V/W (q)), part (3) follows.

This completes the proof of the Proposition.

Theorem 3. Let Hm(n;M) denote Hm(ET(An);M) where M is a Z[1/6]-module. We have:(1) H1(n;M) = 0 for all n > 2,(2) H0(GL3(A), H2(3;M))→ H2(n;M) is an isomorphism for all n ≥ 4,(3) H0(GL4(A), H3(4;M))→ H3(n;M) is an isomorphism for all n ≥ 5,(4) H0(GL2m−2(A), Hm(2m − 2;M)) → Hm(n;M) is an isomorphism for alln > 2m− 2.

Proof. Part (1) has already been proved.Proof of part 2. For this, we study SS(V ;M) where rank(V ) = 4. We firstnote that(i) E2

3,0 = D(V ) and therefore H0(PGL(V ), E23,0) = 0.

(ii) E21,1 = E1

1,1 = ⊕det(q) ⊗ D(V/W (q)) : q ∈ L1(V ), and therefore

H1(PGL(V ), E21,1) = 0 by IIa.

(iii) E2u,v = 0 except when (u, v) = (0, 0), (0, 2), (1, 1), (3, 0). We have E∞

1,1 = 0by proposition 28 and thus obtain the short exact sequence:0→ E3

3,0 → E23,0 → E2

1,1 → 0.

By (i) and (ii) above, we see that H0(PGL(V ), E33,0) = 0. By Ib(1,4), we see

that H0(PGL3(A), H2(3;M)) → H2(4;M) is an isomorphism. In particular,H2(4;M) receives the trivial PGL4(A)-action. Taking N = 2 and m(1) = 0 inlemma 27, we see that H2(4;M)→ H2(n;M) is an isomorphism for all n ≥ 4.This proves part (2).Proof of part 3. We inspect SS(V ;M) where V = A5. We note that(1) E∞

1,2 and E∞2,1 both vanish. This follows from proposition 28, once it is

noted that KH1(2;M) = H1(2;M).(2) E2

0,2 = H2(4;M) has the trival PGL(V )-action.



(3) Hi(PGL(V ), E22,1) = 0 for all i < 3. This follows from IIa and IIIa(3) after

observing that H1(2;M) ∼= D(A2)⊗M .(4) From (2) and (3) we see that d22,1 = 0.

(5) We deduce that E22,1∼= E2

0,4/E30,4 and E2

1,2 = E31,2∼= E3

0,4/E40,4 from obser-

vations (1) and (4).(6) H1(PGL(V ), E2

1,2) = 0.To see this, first note the the short exact sequence:0→ P → E2

1,2 → Q→ 0, whereP = ⊕det(q)⊗KH2(V/W (q) : q ∈ L1(V ) and Q = H2(4;M)⊗ Z1C•(V ).The vanishing of H1(PGL3(A), Q) follows from IIIa(5). By IIa, the vanishingof H1(PGL3(A), P ) is reduced to the vanishing of H0(PGL3(A),KH2(A

3).Now let I be the augmentation ideal of the group algebra R[PGL3(A)] whereR = Z[1/6]. In view of the fact that PGL3(A)ab is 3-torsion, we see thatI = I2. It follows that for all Z[1/6]-modules N equipped with PGL3(A)-action, we have IN = I2N , or equivalently, H0(PGL3(A), IN) = 0. We applythis remark to N = H2(3;M). By part (2) of the proposition, we see thatKH2(3;M) = IN . This proves that H0(PGL3(A),KH2(A

3)) = 0. We havecompleted the proof of observation 6.(7) H0(PGL(V ), E4

4,0) = 0.

In view of the filtration of (5), it suffices to check that H1(PGL(V ), E2a,b) = 0

for (a, b) = (1, 2) and (2, 1) (which has been seen in observations (3) and (6))and also that H0(PGL(V ), E2

4,0) = 0 (and this is clear because E24,0 = D(V )).

(8) H0(PGL4(A), H3(4;M))→ H3(V ;M) is an isomorphism.That H0(PGL4(A), H3(4;M))→ E∞

0,3 is an isomorphism follows from observa-tion (7) and Ib(4). Now E∞

a,b = 0 whenever a+ b = 3 and (a, b) 6= (3, 0). This

proves (8).(9) H3(5;M)→ H3(n;M) is an isomorphism for all n ≥ 5.This follows from lemma 27 by taking N = 3 and m(1) = m(2) = 0. Thisfinishes the proof of part (3).Part (4) now follows from the same lemma and induction.

Remark. It can be checked that parts (1,2,4) of the above theorem are validfor Z[1/2]-modules M . In part (3), it is true that H3(n;M) ∼= H3(n + 1;M)for n > 4 and also that H0(PGL4(A), H3(4;M)→ H3(5;M) is a surjection.

Proposition 29. Assume that the Compatible Homotopy Question has an af-firmative answer. Then, for all Z[1/r!]-modules M and for all d > r + 1,H0(PGLr+1(A), Hr(r + 1;M))→ Hr(d;M) is an isomorphism.

Proof. For r = 1, this statement has been checked in IIIb(3) and lemma 27.Let N > 1. We assume that the above statement has been proved for all r < N .Let M be a Z[1/N !]-module. In lemma 27, may may now take m(1) = m(2) =... = m(N − 1) = 0. From this lemma, we obtain:

H0(PGLN+2(A), HN (N + 2;M))→ HN (N ′;M)



is an isomorphism for all N ′ > N + 2. So the proposition is proved once it ischecked that

H0(PGLN+1(A), HN (N + 1;M))→ HN (N + 2;M)

is an isomorphism. To prove this, we consider the spectral sequence SS(V ;M)where V = AN+2. We will prove:(i) E2

a,b = 0 or a = 0 or a + b = N or (a, b) = (N + 1, 0). Furthermore the

action of PGL(V ) on E20,b is trivial when b < N .

(ii) if a > 0 and b > 0, then Hi(PGL(V ), E2a,b) = 0 for i = 0, 1.

(iii) E∞a,b = 0 when a > 0 and b > 0.

(iv) E2a,b∼= Eb+1

N+1,0/Eb+2N+1,0 whenever a > 0, b > 0 and a+ b = N .

We first observe that (iv) is true for any spectral sequence of PGL(V )-modules where (i),(ii) and (iii) hold. Next note that (ii) and (iv) imply that

H0(PGL(V ), EN+1N+1,0) is contained in H0(PGL(V ), E2

N+1,0). And since thelatter is zero, we see that the former also vanishes.We deduce that both arrowsH0(PGL(V ), E1

0,N )→ E∞0,N → HN (N+2;M) are

isomorphisms exactly as in earlier proofs. Thus it only remains to prove (i),(ii) and (iii).Proof of (i). This is contained in the proof of lemma 27.Proof of (ii). 0→ P → E2

a,b → Q→ 0 is exact, where

P = ⊕det(q) ⊗ KHb(V/W (q)|q ∈ La(V ), and Q = ZaC• ⊗ Hb(b + 2;M)as in the proof of the lemma 27. The required vanishing of Hi(PGL(V ), T )for i = 0, 1 holds for T = Q by IIIa(5). For T = P and a > 1, the requiredvanishing follows from II(a). For T = P and a = 1, this is deduced from thevanishing of H0(PGLN (A),KHN−1) (see the proof of observation (6) in theproof of theorem 3).Proof of (iii). We follow the steps of the proof of Proposition 28. We firstchoose q ∈ LN−1(V ) and consider the inclusion U(q) → ET(V ). As in thatproof we get a homomorphism of E1 spectral sequences of G-modules withG ⊂ SL(V ) as given there. The terms of these spectral sequences are denotedby Ea

b,c(q) and Eab,c(V ) respectively. From the inductive hypothesis, we deduce:

(i′) E2a,b(q) = 0 or a = 0 or a+ b = N or (a, b) = (N + 1, 0). Furthermore the

action of G on E20,b(q) is trivial when b < N .

(ii′) if a > 0, h > 0, then H0(G,Eha,b(q)) = 0.

These observations together imply(iii′) Z∞

a,b(q) = Z2a,b(q) when a > 0 and b > 0.

For a > 0, b > 0, we obtain E∞a,b(q) → E∞

a,b(V ) is zero, from the affirmative

answer to the Compatible Homotopy Question. For such (a, b),the image of x(q) : Z2

a,b(q) → Z2a,b(V ) is thus contained in B∞

a,b(V ). As in

the proof of proposition 28, we see that the sum of the images of x(q), takenover all q ∈ LN−1(V ), is all of Z2

a,b(V ). It follows that Z2a,b(V ) = B∞

a,b(V ) and

thus E∞a,b(V ) = 0 whenever a > 0, b > 0. This proves assertion (iii) and this

completes the proof of the Proposition.



10. a double complex

We will continue to assume that A is a commutative ring with many units. Thepaper [2] of Beilinson, Macpherson and Schechtman introduces a Grassmanncomplex, intersection and projection maps, and a torus action. The terms ofthe double-complex constructed below may be obtained from the quotients bythe torus action of the objects of [2]. The arrows of the double-complex aresigned sums of their intersection and projection maps.D(V ), C•(V ) etc. are as in the previous section. When rank(V ) = n, we havethe resolution:0← D(V )← Cn(V )← Cn+1(V )....We put Cr(V ) = H0(PGL(V ), Cr(V )) when r ≥ n and define Cr(V ) to bezero otherwise. We put Cr(A

n) = Cr(n). We observe that the above res-olution of D(V ) tensored with the rationals is a projective resolution in thecategory of Q[PGL(V )]-modules. It follows that Hi(PGLn(A), D(An))⊗Q ∼=Hn+i(C(n)•)⊗Q. We denote by ∂′ : Cr(n)→ Cr−1(n) the boundary operatorof C(n)•. We will now define ∂′′ : Cr(n)→ Cr(n− 1).Let V ∼= An. Let (L0, L1, ..., Lr) be an ordered (r + 1)-tuple in L(V ) thatgives rise to a r-simplex of K(V ) (see consequence III of many units fornotation). We define ∂i(L0, L1, ..., Lr) ∈ Cr−1(V/Li) by ∂i(L0, L1, ..., Lr) =(L0, .., Li−1, Li+1, ..., Lr) where Lj = Lj + Li/Li ∈ L(V/Li) whenever j 6= i.Now let

gr(L0, L1, ..., Lr) = Σri=0(−1)

i∂(L0, L1, ..., Lr) ∈ ⊕Cr−1(V/L) : L ∈ L(V ).

The above gr : Cr(V ) → ⊕Cr−1(V/L) : L ∈ L(V ) anti-commutes withthe boundary operator. The functor M → H0(PGL(V ),M) takes gr to ∂′′ :Cr(n)→ Cr−1(n− 1). This defines ∂′′.We put Fr(A) = ⊕Cr(n) : n ≥ 1 and define ∂ : Fr(A) → Fr−1(A) by∂ = ∂′ + ∂′′. The exact relation between the homology of F•(A) and groupsLn(A) is as yet unclear. However, we do have:

Lemma 30. H3(F•(A))⊗Q ∼= L2(A) ⊗Q ∼= H3(C•(2))⊗Q.

We sketch a proof. In view of the H-space structure, Li(A)⊗Q is the primitivehomology of ET(An) with Q coefficients for n large. The vanishing of H1(n;Q)for n > 2 implies that the primitive homology is all of Hi(n;Q) for i = 2, 3 andn large. By theorem 3, we get Li(A) ⊗ Q ∼= H0(PGLi+1(A), Hi(i + 1;Q)) fori = 2, 3. For the computation of H0(PGL(V ), H2(ET(V );Q)) where V = A3,we recall the exact sequence obtained from SS(V ;Q):0→ H2(ET(V );Q)→ D(V )⊗Q→ D2(V ) = ⊕D(V/L) : L ∈ L(V ) → 0.This identifies L2(A) ⊗ Q with the cokernel of H1(PGL(V ), D(V )) ⊗ Q →H1(PGL(V ), D2(V ))⊗Q. In view of IIa and the above remarks, this is readilyidentified with H3(F•(A))⊗Q.That gives the first isomorphism of the lemma.For the second isomorphism, what one needs is:Claim: The arrow H4(C•(3))→ H3(C•(2)) induced by ∂′′ is zero.The proof of this claim, which we address now, was already known to SpencerBloch. Let V = A3. Given an ordered 5-tuple (L0, ..., L4) with the Li ∈ L(V )



as vertices of a 4-simplex in K(V ) (i.e. in general position), they belong toa conic C and the projection from the points Li induces an isomorphism pi :C → P(V/Li). We put (M0, ...,M4) = (p0L0, p0L1, ..., p0L4). Let qi = pi p

−10 .

With the ∂i as in the definition of g4, we see that ∂i(L0, ..., L4) ∈ C3(V/Li)and qi∂i(L0, ..., L4) ∈ C3(V/L0) both give rise to the same element of C3(2).It follows that ∂(M0,M1, ...,M4) 7→ ∂′′(L0, ..., L4) under the map C3(V/L0)→C3(2). Thus ∂′′(L0, ..., L4) 7→ 0 ∈ H3(C•(2). This proves the claim and thelemma.Thus we have shown thatL2(A)⊗Q ∼= coker(C4(A

2)→ C3(A2)).

The Bloch group tensored with Q is the homology of

C4(A2)→ C3(A

2)→ Λ2(A×)⊗Q.

Thus this discussion amounts to a proof of Suslin’s theorem on the Bloch group.It remains to obtain a closed form for L3(A)⊗Q by this method.Acknowledgements: This paper owes, most of all, to Sasha Beilinson whointroduced us to the notion of a homotopy point. The formulation in termsof contractible spaces of maps occurs in many places in this manuscript. Robde Jeu lent a patient ear while we groped towards the definition of FL(V ).Peter May showed us a proof of Proposition 20. We thank them, Kottwitzand Gopal Prasad for interesting discussions, and also Spencer Bloch for hissupport throughout the project. Finally we thank the referee who brought ourattention to several careless slips in the first draft.

References

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Birkhauser (1996).[20] A. A. Suslin Homology of GLn, characteristic classes and Milnor K-

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M. V. NoriDept. of MathematicsUniversity of Chicago5734 University [email protected]

V. SrinivasSchool of MathematicsTata Instituteof Fundamental Research

Homi Bhabha [email protected]


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K-Theory and the Enriched Tits Building - uni-bielefeld.de · K-Theory and the Enriched Tits Building To A. A.Suslinwith admiration,on hissixtieth birthday. M. V.Noriand V. Srinivas